AN ELEMENTARY TEXT-BOOK 



ON THE 



DIFFERENTIAL AND INTEGRAL 
CALCULUS 



WILLIAM H. ECHOLS 

Professor of Mathematics in the University of Virginia. 





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NEW YORK 

HENRY HOLT AND COMPANY 

1902 



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Copyright, 1902, 

BY 

EENRY HOLT & CO. 



ROBERT lUII'MMOSR, PRINTER, NEW YORK. 



PREFACE. 

This text-book is designed with special reference to the needs of 
the undergraduate work in mathematics in American Colleges. 

The preparation for it consists in fairly good elementary courses 
in Algebra, Geometry, Trigonometry, and Analytical Geometry. 

The course is intended to cover about one year's work. Experi- 
ence has taught that it is best to confine the attention at first to func- 
tions of only one variable, and to subsequently introduce those of 
two or more. For this reason the text has been divided into two 
books. Great pains have been taken to develop the subject con- 
tinuously, and to make clear the transition from functions of one 
variable to those of more than one. The ideas which lie about the 
fundamental elements of the calculus have been dwelt upon with 
much care and frequent repetition. 

The change of intellectual climate which a student experiences in 
passing from the finite and discrete algebraic notions of his previous 
studies to the transcendental ideas of analysis in which are involved 
the concepts of infinites, infinitesimals, and limits is so marked that 
it is best to ignore, as far as possible on first reading, the abstruse 
features of those philosophical refinements on which repose the foun- 
dations of the transcendental analysis. 

The Calculus is essentially the science of numbers and is but an 
extension of Arithmetic. The inherent difficulties which lie about its 
beginning are not those of the Calculus, but those of Arithmetic and 
the fundamental notions of number. Our elementary algebras are 
beginning now to define more clearly the number system and the 
meaning of the number continuum. This permits a clearer presen- 
tation of the Calculus, than heretofore, to elementary students. 

As an introduction and a connecting link between Algebra and 
the Calculus, an Introduction has been given, presenting in review 
those essential features of Arithmetic and Algebra without which it is 
hopeless to undertake to teach the Calculus, and which are unfor- 
tunately too often omitted from elementary algebras. 

The introduction of a new symbolism is always objectionable. 



IV PREFACE. 

Nevertheless, the use of the " English pound " mark for the symbol 
of "passing to the limit" is so suggestive and characteristic that 
its convenience has induced me to employ it in the text, particularly 
as it has been frequently used for this purpose here and there in the 
mathematical journals. 

The use of the "parenthetical equality" sign ( = ) to mean 
" converging to " has appeared more convenient in writing and print- 
ing, more legible in board work, and more suggestive in meaning than 
the dotted equality, =, which has sometimes been used in American 
texts. 

An equation must express a relation between finite numbers. The 
differentials are denned in finite numbers according to the best mod- 
ern treatment. In order to make clear the distinction between the 
derivative and the differential-quotient, I have at first employed the 
symbol Df, after Arbogast, or the equivalent notation f of Lagrange 
exclusively, until the differential has been defined, and then only has 
Leibnitz's notation been introduced. After this, the symbols are 
used indifferently according to convenience without confusion. 

The word quantity is never used in this text where number is 
meant. True, numbers are quantities, but a special kind of quantity. 
Quantity does not necessarily mean number. 

The word ratio is not used as a relation between numbers. It is 
taken to mean what Euclid defined it to be, a certain relation between 
quantities. The corresponding relation between numbers is in this 
book called a quotient. The quotient of a by b is that ?iumber whose 
product by b is equal to a. 

In preparing this text I have read a number of books on the 
subject in English, French, German, and Italian. The matter pre- 
sented is the common property now of all mankind. The subject 
has been worked up afresh, and the attempt been made to present it 
t<> American students after the best modern methods of continental 
writers. 

I am especially indebted to the following authors from whose 
books the examples and exercises have. been chiefly selected : Tod- 
hunter, Williamson, Price, Courtenay, Osborne, Johnson, Murray, 
Boole, Laurent, Serret, and Frost. 

My thanks are due Dr. John E. Williams for great assistance in 
reading the proof and for working out all of the exercises. 

W. PL E. 

University of Virginia, October, 1902. 



CONTENTS . 

INTRODUCTION. 

FUNDAMENTAL ARITHMETICAL PRINCIPLES. 

PAGE 

Section I. On the Variable. i 

The absolute number. The integer, reciprocal integer, rational num- 
ber. The infinite and infinitesimal number. The real number system 
and the number continuum. Variable and constant. Limit of a vari- 
able. The Principle of Limits. Fundamental theorems on the limit. 

Section II. Function of a Variable 19 

Definition of functionality. Explicit and implicit functions. Continu- 
ity of Function. Geometrical representation. Fundamental theorem of 
continuity. 

BOOK I. 

FUNCTIONS OF ONE VARIABLE. 
PART I. 

PRINCIPLES OF THE DIFFERENTIAL CALCULUS. 
Chapter I. On the Derivative of a Function 35 

Difference of the variable. Difference of the function. Difference- 
quotient of the function. The derivative of the function. Ab initio 
differentiation. 

Chapter II. Rules for Elementary Differentiation 41 

Derivatives of standard functions log x, x a , sin x. Derivative of sum, 
difference, product and quotient of functions in terms of the component 
functions and their derivatives. 

Derivative of function of a function. Derivative of the inverse function. 
Catechism of the standard derivatives. 

Chapter III. On the Differential of a Function 55 

Definition of differential. Differential-quotient. Relation to difference. 
Relation to derivative. 

Chapter IV. On Successive Differentiation 62 

The second derivative. Successive derivatives. Successive differentials. 

The differential-quotients, variable independent. Leibnitz's formula for 

the nth. derivative of the product of two functions. Successive derivatives 
of a function of a linear function of the variable. 



vi CONTENTS. 

PAGE 

Chapter V. On the Theorem of Mean Value 74 

Increasing and decreasing functions. Rolle's theorem. Lagrange's 
form of the Theorem of Mean Value, or Law of the Mean. Cauchy's form 
of the Law of the Mean. 

Chapter VI. On the Expansion of Functions 82 

The power-series. Taylor's series. Maclaurin's series. Expansion of 
the sine, logarithm, and exponential. Expansion of derivative and primi- 
tive from that of function. 

Chapter VII. On Undetermined Forms 92 

Cauchy's theorem. Application to the illusory forms 0/0, 00 /oo , 

o x » , <» - w , o°, 00 °, i 00 . 

Chapter VIII. On Maximum and Minimum 103 

Definition. Necessary condition. Sufficient condition. Study of a 
function at a point at which derivative is o. Conditions for maximum, 
minimum, and inflexion. 



PART II. 

APPLICATIONS TO GEOMETRY. 
Chapter IX. Tangent and Normal 112 

Equation of tangent. Slope and direction of curve at a point. Equa- 
tion of normal. Tangent-length, normal-length, subtangent and subnor- 
mal. Rectangular and polar coordinates. 

Chapter X. Rectilinear Asymptotes 121 

Definitions. Three methods of finding asymptotes to curves. Asymp- 
totes to polar curves. 

Chapter XI. Concavity, Convexity, and Inflexion 127 

Contact of a curve and straight line. Concavity. Convexity. In- 
flexion, concavo-convex, convexo-concave. Conditions for form of curve 
near tangent. 

Chapter XII. Contact and Curvature . 130 

Contact of two curves. Order of contact. Osculation. Osculating circle, 
circle of curvature, radius, and center. 

Chapter XIII. Envelopes 138 

Definition of curve family. Arbitrary parameter. Enveloping curve 
of a family. Envelope tangent to each curve. 

Chapter XIV. Involute and Evolute 144 

Definitions. Two methods of finding evolute. 

Chapter XV. Examples of Curve Tracing 147 

Curve elements. Explicit and implicit equations. Tracings of simple 
curves. 



CONTENTS. vii 

PART III. 

PRINCIPLES OF THE INTEGRAL CALCULUS. 

PAGE 

Chapter XVI. On the Integral of a Function 165 

Definition of element. Definition of integral. Limits of integration. 
Integration tentative. Primitive and derivative. A general theorem on 
integration. The indefinite integral. The fundamental integrals by ab 
initio integration. 

Chapter XVII. The Standard Integrals. Methods of 
Integration 175 

The irreducible form / dn. The catechism of standard integrals. Prin- 



>J 



ciples of integration. Methods of integration. Substitution (transforma- 
tion, rationalization). Decomposition v parts, partial fractions). 

Chapter XVIII. Some General Integrals 193 

Binomial differentials. Reduction by parts. Trigonometric integrals. 
Rational functions. Trigonometric transformations. Rationalization. 
Integration by series. 

Chapter XIX. On Definite Integration 215 

Symbol of substitution. Interchange of limits. New limits for change 
of variable. Decomposition of limits. A theorem of mean value. Exten- 
sion of the Law of Mean Value. The Taylor-Lagrange formula with the 
terminal term a definite integral. Definite integrals evaluated by 
series. 



PART IV. 

APPLICATIONS OF INTEGRATION. 
Chapter XX. On the Areas of Plane Curves 226 

Areas of curves, rectangular coordinates, polar coordinates. Area 
swept over by line segment. Elliott's extension of Holditch's theorem. 

Chapter XXI. On the Lengths of Curves 243 

Definition of curve-length. Length of a curve, rectangular coordinates, 
polar coordinates. Length of arc of evolute. Intrinsic equation of curve. 

Chapter XXII. On the Volumes and Surfaces of Revolutes. 255 

Definition of rotation. Revolute. Volume of revolute. Surface of 
re volute. 

Chapter XXIII. On the Volumes of Solids 264 

Volume of solid as generated by plane sections parallel to a given 
plane. 



viii CONTENTS. 

BOOK II. 
FUNCTIONS OF MORE THAN ONE VARIABLE 

PART V. 

PRINCIPLES AND THEORY OF DIFFERENTIATION. 

PAGE 

Chapter XXIV. The Function of Two Variables 273 

Definition. Geometrical representation. Function of independent 
variables. Function of dependent variables The implicit function of 
several variables. Contour lines. Continuity of a function of two vari- 
ables. The functional neighborhood. 

Chapter XXV. Partial Differentiation of a Function of 

Two Variables 282 

On the partial derivatives. Successive partial differentiation. Theorem 
of the independence of the order of partial differentiation. 

Chapter XXVI. Total Differentiation 290 

Total derivative defined. Total derivative in terms of partial deriva- 
tives. The linear derivative. Total differential. Differentiation of the 
implicit function. 

Chapter XXVII. Successive Total Differentiation 299 

Second total derivative and differential of z = f(x, y). Second deriva- 
tive in an implicit function in terms of partial derivatives. Successive 
total linear derivatives. 

Chapter XXVIII. Differentiation of a Function of Three 
Variables 306 

The total derivative. The second total derivative. Successive linear 
total differentiation. 

Chapter XXIX. Extension of the Law of Mean Value to 
Functions of Two and Three Variables 309 

Chapter XXX. Maximum and Minimum. Functions of Sev- 
eral Variables 314 

Definition. Conditions for maxima and minima values of f{x , y) and 
f{x. y, z). Maxima and minima values for the implicit function. Use of 
Lagrange's method of arbitrary-multipliers. 

Chapter XXXI. Application to Plane Curves 329 

Definition of ordinary point. Equations of tangent and normal at an 
ordinary point. The inflectional tangent, points of inflexion. Singular 
point. Double point. Node, conjugate, cusp-conjugate conditions. 
Triple point. Equations of tangents at singular points. Homogeneous 
coordinates. Curve tracing. Newton's Analytical Polygon, for separat- 
ing the branches at a singular point. Envelopes of curves with several 
parameters subject to conditions. Use of arbitrary multipliers. 



CONTENTS. IX 

PART VI. 

APPLICATION TO SURFACES. 

PAGB 

Chapter XXXII. Study of the Form of a Surface at a Point. 347 

Review of geometrical notions. General definition of a surface. Gen- 
eral equation of a surface. Tangent line to a surface. Tangent plane 
to a surface. Definition of ordinary point. Inflexional tangents at an 
ordinary point. Normal to a surface. Study of the form of a surface 
at an ordinary point, with respect to tangent plane, with respect to 
osculating conicoid. The indicatrix. Singular points on surfaces. 
Tangent cone. Singular tangent plane. 

Chapter XXXIII. Curvature of Surfaces 365 

Normal sections. Radius of curvature. Principal radii of curvature. 
Meunier's theorem. Umbilics. Measures of curvature of a surface. 
Gauss' theorem. 

Chapter XXXIV. Curves in Space 375 

General equations of curve. Tangent to a curve at a point. Oscu- 
lating plane. Equations of the principal normal. The binormal. Circle 
of curvature. Tortuosity, measure of twist. Spherical curvature. 

Chapter XXXV. Envelopes of Surfaces 385 

Envelope of surface-family having one arbitrary parameter. The 
characteristic line. Envelope of surface-family having two independent 
arbitrary parameters. Use of arbitrary multipliers. 

PART VII. 

INTEGRATION FOR MORE THAN ONE VARIABLE. MULTIPLE 

INTEGRALS. 

Chapter XXXVI. Differentiation and Integration of In- 
tegrals 391 

Differentiation under the integral sign for indefinite and definite inte- 
grals. Integration under the integral sign for indefinite and definite 
integrals. 

Chapter XXXVII. Application of Double and Triple 
Integrals 396 

Plane Areas, double integration, rectangular and polar coordinates. 
Volumes of solids, double and triple integration, rectangular and polar 
coordinates. Mixed coordinates. Surface area of solids. Lengths of 
curves in space. 

Chapter XXXVIII. Differential Equations of First Order 
and Degree 409 

Rules for solution. Exact and non-exact equations. Integrating 
factors. 

Chapter XXXIX. Examples of Differential Equations of 
the First Order and Second Degree 428 

Rules for solution. Orthogonal trajectories. The singular solution. 
c- and /-discriminant relations. Redundant factors not solutions. Node, 
cusp, and tac- loci. 



CONTENTS. 



Chapter XL. Examples of Differential Equations of the 
Second Order and First Degree 439 

The five degenerate forms. The linear equation, and homogeneous 
linear equation having second member o. 



APPENDIX. 

Note 1. Weierstrass's Example of a Continuous Function which has no deter- 

minate derivative , 45 1 

Note 2. Geometrical Picture of a Function of a Function 453 

Note 3. The nth Derivative of the Quotient of Two Functions 454 

Note 4. The nth Derivative of a Function of a Function 455 

Note 5. The Derivatives of a Function are infinite at the same points at which 

the Function is infinite. 456 

Note 6. On the Expansion of Functions by Taylor's Series 457 

Note 7. Supplement to Note 6. Complex Variable 465 

Note 8. Supplement to Note 6. Pringsheim's Example of a Function for which 
the Maclaurin's series is absolutely convergent and yet the Function and 
series are different 467 

Note 9. Riemann's Existence Theorem. Proof that a one-valued and con- 
tinuous function is integrable 468 

Note 10. Reduction formulae for integrating the binomial differential 470 

Note 11. Proof that a curve lies between the chord and tangent, when the 

chord is taken short enough 471 

Note 12. Proof of the properties of Newton's analytical polygon for curve - 

tracing 47 l 

Index 475 



INTRODUCTION TO THE CALCULUS. 



SECTION I. 
ON THE VARIABLE. 

1. Calculus, like Arithmetic and Algebra, has for its object the 
investigation of the relations of Numbers. It is necessary to under- 
stand that the symbols employed in Analysis either represent numbers 
or operations performed on numbers. 

2. The Symbols. 

i, 2, 3, 4, . . . (i) 

are symbols used to represent the groups of marks which we call inte- 
gers. Thus * 

1=1, 

2 = 1+1, 

3 = 14-1 + 1, 



The system of integers (i) extends indefinitely toward the right, 
as indicated by the sign of continuation. This system is called the 
table of integers. Each integer has its assigned place, once and for 
all, in the table. Any integer in the table is, conventionally, said to 
be greater than any other integer to the left of it, and less than any 
integer to the right of it in the table (i). 

3. Definition of Infinite Integer. — When an integer is so great 
that its place in the table of integers cannot be assigned in such a 
manner that it can be uniquely distinguished from each and every 
other integer, that integer is said to be unassignably great or infinite. 
Mathematical infinity has no further or deeper meaning than this. 

4. The Inverse Integer. — The reciprocals of the integers 

i»Uf ( H ) 

constitute an extension of the table (i) to the left of the integer 1, 
which number is its own reciprocal. As before, any number in this 
table is said to be greater than any number to the left of it, and less 
than any number to the right of it. 

Corresponding to each number in (i) there is a number in (ii), 

* The symbol = is to be read, ' ' is identical with, " or ' ' is the same as. ' ' 

1 



2 INTRODUCTION TO THE CALCULUS. [Sec. I. 

and conversely. Those numbers in (ii) which are the reciprocals of 
the infinite or unassignably great integers, are said to be infinitesimals 
or unassignably small.* 

5. The Absolute Number. The Absolute-Number Continuum. 

When in the table of numbers 

. . . , |, 1 1, 2, 3, . . . (iii) 

the gap between each pair of consecutive numbers is filled in with all 
the rational (fractional) and irrational numbers that are greater than 
the lesser and less than the greater of the pair, we construct a table of 
numbers which is called the absolute-number continuum. Each number 
in this system has its assigned place. It is said to be greater than any 
number to the left of it and less than any number to the right of it. 
Each number in the absolute-number continuum is called an absolute 
number. 

Any and all numbers in the table that are greater than any integer 
that can be uniquely assigned, as in § 3, are said to be infinite or unas- 
signably great. In like manner any number in the table that is less 
than any reciprocal-integer that can be uniquely assigned a place in 
the table is infinitesimal or unassignably small. 

The absolute continuum is thus divided into two classes of num- 
bers : the uniquely assigned or simply the assigned numbers, which we 
call the finite numbers ; and the numbers which cannot be uniquely 
assigned or transfinite numbers. 

The transfinite numbers greater than 1 are called infinite, those less 
than 1 infinitesimal numbers. 

6. Zero and Omega. — The absolute-number system, as con- 
structed in § 5, extends indefinitely both ways, in the direction of the 
indefinitely great and in that of the indefinitely small. In this sys- 
tem there is no number greater than all other numbers in' the system, 
nor is there any number that is less than all others in the system. 

The system is conventionally closed on the left by assigning in the 
table a number zero whose symbol is o, which shall be less than any 
number in the absolute system. Since now there is no number greater 
than 1 to correspond to the reciprocal of this number o, we design 
arbitrarily a number omega whose symbol is £1 as the reciprocal of o, 
and which is greater than any number in the absolute system. 

The number o is the familiar naught of Arithmetic. The num- 
ber XI is the ultimate number of the Theory of Functions, and with 
which we shall not be further concerned in this book. 

The number o is not an absolute number, but is the inferior boundary 
number of that system. In like manner the number CI is not an abso- 
lute number, but is the superior boundary number of the absolute 
system. 

* The words ' great ' and ' small ' have in no sense whatever a magnitude mean- 
ing when applied to numbers. They are mere conventional phrases and the words 
1 right ' and ' left 'or ' in ' and ; out,' might just as well be employed. 



Art. 7.] ON THE VARIABLE. 3 

7. The conventional symbol for the whole class of unassignably 
great or infinite numbers is 00 . There has been adopted no conven- 
tional symbol for the class of infinitesimals ; the symbol most com- 
monly used is the Greek letter iota, 1. 

8. The Real-Number System. — When in the algebraic system 
of numbers 

— n, . . . , — 3, — 2, — 1, o, + 1, + 2, + 3, . . . , -f n, 

the gap between each consecutive pair is filled in with all the rational 
and irrational numbers that are greater than the lesser and less than 
the greater of the pair, the system thus constructed is called the real- 
number continuum. 

It is understood that any number in this table is greater than any 
number to the left of it and less than any number to the right of it. 

The modulus of any real number is its arithmetical or absolute 
value. Thus, the modulus or absolute value of -f- 3 or — 3 is the 
absolute number 3. If we employ the symbol x to represent any 
number in the real continuum, then its modulus or absolute value is 
represented by \x\ or mod x. 

In this book we shall be directly concerned only with real num- 
bers and their absolute values. Hereafter when we speak of a number, 
we mean a real number unless otherwise specially mentioned.* 

9. Geometrical Picture of the Real-Number System. — We assume a cor- 
respondence between the points on a straight line and the numbers in the real 
continuum. 

-4 -3 x'-2 -1 o + i +% x±3 44 

(-; h-. I . I , H 1 1 1 1 1 1 

D C P B A. O A B P C D 

Fig. 1. 

Select any point O on a straight line. Choose arbitrarily any unit length ; with 
which construct a scale of equal parts, A, B, C, . . . starting at O proceeding 

* The real-number continuum is a closed system of numbers to all operations 
save that of the extraction of roots. When we consider the square root of a nega- 
tive number we introduce a new number. The complex or complete number of 
analysis is 

x + iy, 
where x and y are any two real numbers, and i is a conventional symbol represent- 
ing -f- V — 1. Corresponding to any real number y there are as many complex 
numbers as there are real numbers x ; and corresponding to any real value x 
there are as many complex numbers as there are real numbers y. The complex 
system is a double system. In the theory of functions of complex numbers, 
which includes that of real numbers as a special case, the ultimate number fl is 
conventionally a number common to all systems in the same way as is o. 

The student is already familiar with the impossibility of solving all questions 
in analysis with real numbers only. For example, in the theory of equations 
when seeking the roots of equations. All the more so is this true in the Calculus, 
for we cannot solve the fundamental problem of expanding functions in series with- 
out the use of complex numbers, except in a very few particular cases. 

If z is any complex number x + iy, its modulus or absolute value is 



4 • INTRODUCTION TO THE CALCULUS. [Sec. I. 

toward the right, and A', £•', C' f . . . toward the left. Mark the points of divi- 
sion, o at the origin O, and -f- I, -j- 2, etc., toward the right ; — I, — 2, etc., 
toward the left. Then, corresponding to any real number x there is a point P oix 
the line to the right of O if x is positive ; and P' to the left of O if x is negative. 
The number x is the measure of the length OP with respect to the unit length 
chosen. Conversely, corresponding to each point Pon the line there is a number 
in the real-number system. 

io. Variable and Constant. — In the continuous number system, 
as designed in § 8, it is convenient to use letters as general symbols 
to represent temporarily the numbers in that system. Thus, we 
can think of a symbol x as representing any particular number in that 
system. Further, we can think of a symbol x as representing any 
particular number, say -\- 3, and then representing continuously in 
succession every number between -j- 3 and any other number, say 
-j- 5, and finally attaining the value + 5. We speak of such a symbol 
x, representing successively different numbers, as a number, and 
w.e speak of any particular number which it represents, as its 
value. 

Definition. — A number x is said to be variable or constant ac- 
cording as it does or does not change its value during an investigation 
concerning it. 

We shall frequently be concerned with symbols of numbers which 
are variable during part of an investigation and are constant during 
another part. 

Generally, variables are represented by the terminal letters u y v, 
w, x,j', z, etc., and constants by the initial letters a, b, c, etc., of 
the alphabet. This is not always the case, however, as the context 
will show. 

11. Interval of a Variable. — We shall sometimes confine our 
attention to a portion of the number system. For example, we may 
wish to consider only those numbers between a and b. We shall employ 
the symbol (a, b), a being less than b, to represent the numbers a, b 
and all numbers between them. If we wish to exclude from this 
system b only, we write (a, b ( ; if a only, we write ) <z, b) ; when we 
wish to exclude a and b and consider only those numbers greater than 
a and less than b, we represent the system by )a, b[. 

If x is a general symbol representing any number in such a portion 
of the number system, or interval, defined by a and b, we have the 
equivalent notations, 

(a, b) = a s. x ^ b, 

(a, 6(=a ±x<b, 

)a, b)~a < x ±b, 

)a, b(=a < x < b. 

A variable x is said to vary continuously through an interval (a, b), 
when x starts with the value a and increases to b in such a manner as 
10 pass through the value of each and every number in (a, b). Or, x 



Art. 12.] ON THE VARIABLE. 5 

passes in like manner from b to a. The number x is said to increase 
continuously from a to b, or decrease continuously from b to a. 






a x 


b 





A P 

Fig. 2. 


B 



In the geometrical picture, of § 9, illustrating the number system, if the points 
A, J\ B, correspond to the numbers a, x, b, the segment AB represents the inter- 
val (a, b). As x varies continuously from a to b, or through the interval {a, b), the 
point /'corresponding to the number x generates the segment AB. 

12. The Limit of a Variable. 

Definition. — When the successive values of a variable x approach 
nearer and nearer to the value of an assigned constant number a in such 
a manner that the absolute value of the difference x — a becomes and 
remains less than any given assigned constant absolute number e what- 
ever, we say that the number x has a for its limit. 

In symbols, under the above conditions we write 

£<*) = * 

which is read, " the limit of x is a. " The variable is said to converge 
to its limit. 

EXAMPLES. 

Arithmetic furnishes examples of a limit . 

1. In the extraction of roots of numbers. Whenever a number has no rational 
number for a root, its root, if real, is an irrational number called a surd, which is 
the limit of a sequence of rational numbers constructed according to a certain law. 

2. In general, the definition of a number is : * 
Any sequence of rational numbers 

a 1 , a 2 , . . . , a M , . . . 
defines and assigns a number, when it is constructed according to any law which 
requires each number in the sequence to be finite and such that, whatever assigned 
number e be given (however small), we can always assign an integer n for which 

\a n — a n+p \ < e, 

for any assigned value of the integer/ (however great). 

The very definition of a number, on which all analysis is founded, is a limit. 
The number assigned by the above regular sequence of numbers is but the limit to 
which converges the element a r of that sequence, as r increases indefinitely. If a 
be the symbol of the number thus defined, then in symbols we write, 

<x= £(a r ). 

It should be observed that irrational numbers having been thus defined, the 
numbers a r in a regular sequence can be any numbers rational or irrational. The 
regular sequence defines and assigns a number in its place in the table of numbers. 

Algebra furnishes a useful and an interesting example of a limit in the evalua- 
tion of the infinite geometrical progression. 

3. The identity 

i_ x* + *=(i — x)(i-{-x-\- x*-\- . . . +x*) 

is established by multiplying the two factors on the right. 
* Due to Cantor and Weierstrass. 



INTRODUCTION TO THE CALCULUS. 



[Sec. L 



Therefore, in compact symbolism, which we shall frequently employ, 



2xr 



X* + 



If x is any number such that \x\ < I. we can make and keep x* + », and there- 
fore also the second term of the member on the right, less than any assigned 
number e, by making n sufficiently great. Therefore, the limit of the sum of the 
series on the left is i/^i — x), or in symbols 

£ i *r= i. + *+■*»+ . . . 



I — x 

In this example the variable is 

S r =I + X + . . . +xr. 

If now x is any assigned number in )o, l(, x is positive, and 6V continuously 
increases as r increases. The variable S r is always less than the limit. If x is in 
)— i, o(, it is negative, say x^£ — a ; then 



^(-i)^ 



( - !)««», 



+ 



(— l)»a«+i 



1+0 

When n is even the variable S n is greater than its limit ; when n is odd 
the variable S n is less than its limit. Therefore, ^s n increases through integral 
values, the variable converges to its limit, changing from greater than the limit to 
less as n changes from even to odd and vice versa. 

It is to be observed that if \x\ > I, the sum of the series and the equivalent 
member on the right increase indefinitely with «, in absolute value, and can be 
made greater than any assigned number and therefore become infinite. Under 
these circumstances the series has no limit ; its value becomes indeterminately 
great. 

Geometry furnishes numerous illustrations of the limit. The most notable being : 

4. The evaluation of the area of the circle as the limit to which converge the 
areas of the circumscribed and inscribed regular polygons as the number of sides 
is indefinitely increased. 

5. The evaluation of the irrational and transcendental number n representing 
the ratio of the circumference of a circle to its diameter. 

Trigonometry furnishes an illustration of a limit which will be found useful 
later: 

6. To evaluate the limit of the quotient sin x ~- x as x diminishes indefinitely 

\ in absolute value. 

Draw a circle with radius I. Draw MA = 
MB perpendicular to OT. Then 

Area quadrilateral OA TB = tan x, 
Area triangle OAMB = sin x, 

Area sector OANB = x, 

where x is, of course, the circular measure of 
Z AOT. 

Then, obviously, from geometrical consider- 
ations, 

sin x < x < tan x, 
x I 

or i < —. — < , 




I > 



sin x cos x 



> cos x. 



Art. 13.] ON THE VARIABLE. 7 

When ,r diminishes" indefinitely in absolute value, cosjc becomes more and more 
nearly equal to I, and has the limit I as x converges to o. Consequently the quo- 
tient (sin x)/x converges to the limit I as .r converges to o. In our symbolism, 



/< 



)= 



13. Definition. — When a symbol x, representing a variable num- 
ber, has become and subsequently remains always less, in absolute 
value, than any arbitrarily small assigned absolute number, x is said 
to be infinitesimal. 

When a variable becomes and remains greater, in absolute value, 
than any arbitrarily great assigned number, the variable is said, to be 
infinite. 

When a variable x is infinitesimal, we write* x( = )o. It follows 
from the definition that when a variable becomes infinitesimal it has 
the limit o, or assigns the number o. 

When x has the limit a, or £x — #, then by definition 

£{x - a) = o. 
When x — a is infinitesimal, we write 
x — a( = )o. 
This same relation we shall frequently express by the symbol 

x( = )a, . . , 

meaning that the absolute value of the difference between x and a is 
infinitesimal. When a is the limit of x, the symbol x( = )a is to be 
read, "as x converges to tf," or " x converges to a." 

We shall frequently use the symbol e (epsilon) to represent an 
arbitrarily small assigned absolute number. We then speak of the 
interval (a — e, a -j- e) as the neighborhood of an assigned number 
a. The symbol x(=)a means that " x is in the neighborhood of a." 
All numbers that are in the neighborhood of an assigned number are 
said to be consecutive numbers. 

When a variable x becomes infinite we write x = 00 . Such a 
variable has no limit, it simply becomes indeterminately great. The 
symbol x = 00 merely means that x is some number in the class of 
unassignably great numbers. 

14. The Principle of Limits. 

I. A variable cannot simultaneously converge to two different 
limits. 

*The equality sign in parenthesis (=) may be read " parenthetically equal 
to," the word ' parenthetically ' carrying with it the explanation of the nature of the 
approximate equality. It is simply another way of saying that the difference 
between two numbers is infinitesimal. 

\x — a\ = 1 and x — a( = )o 

mean the same thing. The symbol = has been used for (=), but appears less con- 
venient, expressive, and explicit. 



8 INTRODUCTION TO THE CALCULUS. [Sec. I. 

It is impossible for a (one-valued) variable x to converge to two 
unequal limits a and b. For, the differences \x — a\ and \x — b\ can- 
not each be less than the assigned constant number \ \ b — a | for the 
same value of x . 

The direct proof of this statement rests on this: 
The number x must be either greater than, equal to, or less than 
the number \{a -\- b), where say a < b. 

If x = ±(a + b), .'. x-a=-$(b-a). 

If x > \{a + t>), .-. x - a > \{b - a). 

If x <i(fl + ^), . •. b - .v > J(* - a). 

II. If two variables x and jy are always equal and each converges to 
a limit, then the limits are equal. 

If £x = a, and £y = b, and x =y = z, then, by I, the variable 
z cannot converge to two unequal limits simultaneously. Therefore 
a-b. 

15. Theorems on the Limit.* 

I. If the limit of x is o, then also the limit of ex is o, where c is 
finite and constant. 

For, whatever be the assigned constant absolute number e, we can 
by definition of a limit make and keep \x\ less than the constant 
I e/c I , and therefore ex less than e in absolute value. Consequently, 
by definition 

£ (ex) = o = c£(x). 

x(=)o 

II. If each of & finite f number of variables x lf x % , . . . , x nf has 
the limit o, then the algebraic sum of these variables has the limit o. 

Let x be the greatest, in absolute value, of the n variables. Then 

l*i + * 2 + • • • +Xn\^nx- 
Since n is finite, the limit of this sum is o, by I. 

III. If £x l = a x , £x 2 = a 2 , . . . , £x n = a H , then when n is a 
finite integer 

£(x x + x 2 + . . . + x H ) = £x x + £x % -f . . . -f £x u . 
For, put x x = a x -f- a y , . . . , x H = a n -\- a H . By definition, the 
limits of a x , . . . , a n are o. Hence 

* t + *,+ ... + *« = fo +...+«.) + K + .... + ««), 
by II, gives 

£(x, + ...+x n ) = a l + ...+* H . 

Therefore the limit of the sum of a finite number of variables is 
equal to the sum of their limits. 

* The theorems of this article are of such fundamental importance and so 
absolutely necessary for the foundation of the Calculus that it will, in general, be 
assumed hereafter that they are so well known as to require no further reference to 
them. 

f If the number of variables is not finite, this theorem does not hold in general. 



Art. 15.] ON THE VARIABLE. <) 

IV. The limit of the product of two variables x x and x 2 which 
have assigned limits a x and a 2 , is equal to the product of their limits. 

Let, as in III, x x = a x + a x , x 2 = a 2 -f a r 

. •. Xy x % = a x a 2 + a x a 2 -f- a 2 a x -f a x a r 
By III, we have 

£(*!**) = «A + «u£" a + * 2 ;£"i + ;£K",)- 
But, ^^ = o, ;£ar 2 = o, and a fortiori £(a x a 2 ) — o. Therefore 
£{x x x 2 ) = a x a 2 = (£x x )(£x 2 ). 

Cor. The limit of the product of & finite number of variables having 
assigned limits, is equal to the product of their limits. In symbols * 

£ n(x r ) = n£(x r ). 

r — 1 r = 1 

V. The limit of the quotient, x x /x 2 , of two variables is equal to 
the quotient of their limits, provided the limit of the denominator is 
not o. 

With the same symbolism as in IV, 

*\ _ a x + a i __ a i 1 a i + a 2 a i 
x 2 <* 2 -{-<* 2 a 2 a 2 +a 2 a 9 * 

a 2 a A a t + <**)' 

By hypothesis, £a x = o, £a 2 = o, and a 2 =£ o. Therefore the 
denominator of the second term on the right is always finite, while, 
by III, the limit of the numerator is o. The limit of this term is 
o, by If 

fx i\ - a _i- i^L 
~ — „ ~ /-„ * 



It 



a 2 £x 2 



VI. If x and y are two variables and a is a constant, such that y 
always lies between x and a, then if £x = a, also £y = a. 

* As the symbol ~2 is used to indicate the sum, so U is used to indicate the 
product of a set of numbers. Thus, 

n 
2x r = X x -f *„ + . . . + X n , 



IIx r rJ;,X^X--- X x n . 

1 

The advantage of such symbolism is in compactness of the formulae. 

t Notice particularly the provision that £x 2 =£ o. For, when £x 2 = o and 
£x x =± o, the quotient Xx/x? inci eases beyond all limit or becomes infinite as x { 
and x 7 converge to their limits. An infinite number cannot be a limit under the 
definition. 

Again, if £x 2 = o and also £x x = o, the quotient of the limits 0/0 is com- 
pletely indeterminate, while the quotient x,/x 2 = q mayor may not converge to 
a determinate limit. The value of this quotient as x, and x 2 converge to o depends 
on the law connecting the variables x } and x 2 as they converge to o. This case is 
one of profound importance and is the foundation of the Differential Calculus. 



io INTRODUCTION TO THE CALCULUS. [Sec. I. 

The truth of this is obvious, since \x — a\>\y — a\, and x — a 
has the limit o. 

In like manner, it follows that if x and z have the common limit 
a, and y is a third variable between x and z, then also must jQy = a. 
For, \y — a\ must at all times be less than one or the other of the 
differences \x — a\ and \z — a\, and each of these differences has 
the limit o. 

VII. If one of two variables is always positive and the other is 
always negative, and they have a common limit, that limit is o. 

Let a be the common limit of x and y, where x is always positive 
and y is always negative. Then 

-f|*| = fl + tf, and —\j>\ = a + /3, 
where £a = o, £(3 = o. Subtracting, 

\x\ + \y\ = <*-p. 

Since £(a — fi) = o, . •. a -f- a = 2a = o, and a, the com- 
mon limit of x and_y, is o. 

VIII. If a variable x continually increases and assumes a value 
a but is never greater than a given constant A, then there must exist 
a superior limit of x equal to or less than A. 

(1). No number such as a which x once attains can be a limit of 
x. For, since x continually increases, it must subsequently take some 
value a' > a, and it is never possible thereafter for x — a to be less 
than the constant a' — a. 

(2). The variable x cannot attain the number A, since if it did, x 
continually increasing must become greater than A, which is contrary 
to hypothesis. 

(3). Divide the interval^ — a = h into 10 equal parts. The vari- 
able x after attaining a must either attain a -f- -^h or remain always 
less than a -f- ^h. \{x attains a -f- ^h, it must either attain a -\- -f^h 
or remain always less than a -f- -f^h. We continue to reason thus 

until we find a digit p x such that x must attain a -| -k and remain 

always less than a -}- <J-^ — h. That is, x must enter and always 

remain in one of the 10 intervals. 

In like manner, divide the interval 



V 10 



10 / 



into 10 equal parts. In the same way we find that x must enter and 
always remain in one of these intervals, and that there is a digit p 2 
such that 

IO IO 2 TO IO 2 IO- 



Art. 15. J ON THE VARIABLE. II 

In like manner, continue this process n times. Then 

z^ io r Z^ io io 

This process can be carried on indefinitely. Consequently the 
construction leads to the constant number 



00 
a — a-\-h\ ** r , 



from which x can be made to differ by a number less than h/io n which 
can be made and kept less than any given number e, for all values of 
n greater than m, where h/io m < e. 

Therefore the constant a is the limit of x, and is either equal to 
or less than A. 

In the same way, we prove the theorem : If a variable x always 
diminishes and attains a value a, but is never less than an assigned 
constant number A, then the variable x has an inferior limit that is 
equal to or greater than A. 

IX. If there be two variables x andj^, such that y is always greater 
than x, and if x continually increases andjy continually decreases, and 
the difference y — x becomes less in absolute value than any assigned 
absolute number e, then there is a constant number greater than x 
and less than jy which is the common limit of x and y. 

By Theorem VIII, x has a superior limit a, andy has an inferior limit 
b. For, any particular value jy x of y fixes a constant than which x 
cannot be greater, and any particular value x x of x fixes a constant 
than which ^cannot be less. Hence, if we put 

x = a — a, and y = b -(- fi, 
we have 

y - x = (a - b) - (a + /?). 

But, £ (y — x) = o, £(a -\- /3) = o; . •. a — b = o. This defines 
the equality of a and b. Therefore x and y converge to a common 
limit. 



/ 



I 2 INTRODUCTION TO THE CALCULUS. [Sec. I. 



EXERCISES. 

1. The successive powers of any assigned number greater than I increase 
indefinitely and become infinite as the exponent becomes infinite. 

Let a be any absolute number, and m any integer. 

Then (i + <x) m > * + ma. (I) 

In fact, (i + af - I -f 2a -f- a 2 > I + 2a. 

The formula (i) is true when m = 2. Assume it to be true when m = n. Then 

( i -\- a) n > I -j- na. 
Multiply both sides by I -f <*■- 

.-. (I + a)"+» > i -f (» -f i)a + «a 2 , 
> i + (»+i)a. 
(I) is true also for » -\- I. But, being true for m = 2, it is also true for m = 3, 
and therefore for m =■ 4, etc., and generally. Therefore, since ma and conse- 
quently (1 -\- a) m can be made greater than any assigned number, the proposition 
is demonstrated. 

2. The successive powers of any assigned absolute number less than 1 diminish 
indefinitely and have o for limit. 

Any number less than I can be written as the quotient 1/(1 -f- a). By Ex. 1, 
11 I 

(i-f- a) m i-\-ma ma' 
This can be made less than any assigned number e, by sufficiently increasing m. 

3. The successive roots of an absolute number greater than 1 continually 
diminish ; those of an absolute number less than I continually increase ; and in 
either case have the limit I. 

Whatever be the absolute number a, 



- (*+ 

a n = a 



D_L_ f ^_J* +I 



Therefore, by Exs. 1, 2, whatever be the integer n, 







a n > a n - l , if a > 1 ; 






a n < a M+I , is a < I. 


Ua> 1, 


then 


a" > 1. 


Let 




a — 1 -f- «-, and a w ~ 


then 




(1 + a)« = 1 + A 


or 




(i4>a) = (i-(-/J)«> i+« 
/? < a/«, and we have 



-1+/? 



Hence ^ a « 



Art. 15.] EXERCISES. 13 

Let a < 1, say a =1/(1 -f a). 

1 1 

Also, a n <i, say a H = 1 /( 1 -f- ft). 
Then, as before, ft < a/n t and 

1 
1 > a*> 



i + a/n' 
which shows again that 

£ a« = i. 

4. Show that when a is any assigned positive number, 

£«* = h 

whatever be the way in which x converges to o. 

(1). Let m, n, p, q be any positive integers. Then 

tn p m p 

a n a<f = a M 1 . 



,r_ ™ — f — -f A\ —zL 

a \n q J < a w< 



If a > I, then a* > I 

t 
a n • q > a n f an( i 

Therefore a x continually increases as x increases by rational numbers. 
P_ 
If a < 1, then ai < I. 

— + -£- — — ^^L + i.^ _!!L 

. •. a* 1 < a n , and a V » ■//># ». 

Therefore a* continually diminishes as x increases by rational numbers. 

When \x\ is rational and less than I, there can always be assigned two con- 
secutive integers m and m -\~ 1 such that 

r— < 1*1 < -• 

tn -f- I tn 

The above results show that whether a be greater or less than 1, a* lies between 



a™ + l and a m . When m .— 00 , a m + l and a m converge to I, Ex. 3, and there- 
fore also does a x \ and £a x =r 1, when x(=)o. 

(2). When x is irrational there can always be assigned two rational numbers a 
and ft differing from each other as little as we choose, such that a < x < ft. 
The number a x is defined by its lying between a a and aP. Since x( = )o when 
a( =)ft(= )o, we have, as before, a x converging to I along with a a and aP. 

5. Show that £a* = a £x = aP, if £x = ft. 

We have a& — a x = a&{ I — a x - &). 

Passing to limits, we have, by Ex. 4, 

aP — £a x — o. 

6. If a and ft are positive numbers, and £x — ft, show that 

£ log* x = log a £(x) — log a /?. 

We have log a ft — log a x = log a — . 

The above exercises show that however x converges to ft, £ \og a {ft/x) = o. 
Therefore 

loga P ~ £ ^ga X = O. 



14 INTRODUCTION TO THE CALCULUS. [Sec. I. 

7. Utilize Ex. 6, to prove IV. V, from III, § 15. 

8. Use Ex. 6, to show that 

£(y*) = (£)>) £x . 

where y has a positive limit, and the limit of x is determinate. 

9. A set of numbers a t , a 2 , . . . , a r , . . . , arranged in order is called a se- 
quence. Any number of the sequence, a r , is called an element of the sequence ; the 
number r is called the order of the element a r . Any sequencers said to be known 
when each element is finite and known when its order is known. 

If a lt a 2 , . . . , a H , . . . be a sequence of numbers such that a r is finite when 
r is finite, then will £a ny when n = 00 , be o or 00 according as 



£\ 






is less or greater than 1, respectively. 

Let, when n — oo , £(&n + i/«») — k, and k > 1. Then, by the definition of a 
limit, we can always assign a number k' such that I < k? < k, whence corre- 
sponding to W we can find an integer m for which we have, for all values of «, 

On ±J1±1 > p 

a n + m 

••• a,n + 1 > & a m , 

&m + 2 > k' a m + j > k a m , 

a m + n> k' n a m . 
By hypothesis, a m is finite. Since we can make k' n greater than any assigned 
number by sufficiently increasing n, we have £a n = 00 . 
In like manner, if £{a n + Ja r ?) = k < I, 

a m + H < k' n a m , 
which can be made less than any assigned number by increasing n, when as before 
I > k' < k. . \ £ci n = o, when n = 00 . 

In order that the element a n may have a finite-limit different from o, it is neces- 
sary that * 

I | = | I. 



/' 



The quotient, a n + 1 /a M> of each element by the preceding one will hereafter 
be called the convergency quotient of the sequence. This theorem is of importance 
and will be used later. 

10. The series of numbers 

a i + a 2 +•..+««+•• • (i) 

is said to be absolutely convergent when the corresponding series of the absolute 
values of the terms is convergent. 
That is, when 

5W=|«xl + l«al+ • • • +l*«l 
has a determinate limit when n = 00 . 

Show that (i) is absolutely convergent if 



/ 






and if this limit is greater than 1, the sum of the series is 00 . 

* When the symbols | = |, |>|, |<j are used, they mean that the equality or in- 
equality of the absolute values of the two members of the equation is asserted. 



ART. 15] EXERCISES. 15 

Let the letters in (i) represent absolute numbers, and let 



£ 



Then there can always be assigned an integer m corresponding to any number 
k' such that k < k' < 1, for which 

a m+ n +1 < ^ 



a m+n 

for all values of n. As in Ex. 9, we have 

a>n+n < k' n a m . 
Hence the sum of the series after a m is less than 

k'a m + . . . + U«a m + . . . =a m (k>+ . . . + £'* + ■ ■ • ), 

k' 



This is finite, since k' ^ I. Therefore S x must be finite. Also, by Ex. 9, 
£ a m = o, when m = 00 . Consequently we can always assign an integer n such 
that 

for all values of m, where e is any assigned number. Hence S n has a determi- 
nate limit. Otherwise, the existence of the limit of S tt follows at once from VIII, 
§15. For ^continually increases, but can never exceed 

k' 
*! + *>+■'••+ a ™ + am YZTk>' 

Again, if £(a n +j/a n ) > 1, say equal to k > I. Then, as before, we can assign 
k' between k and 1, and have the sum of the series after a m greater than 

a m (#+...+#«+...), 

which is 00 . 

The number £(a n ±\[an) is called the convergency quotient of the series. 

11. The arithmetical average, or mean value of a sequence of n numbers, 

a \y <*%, ' ' ' t a n, . 

is one nth. of their sum, or 

1 H 
n 

Show that when the number of elements in a sequence increases indefinitely 
according to any given law, the mean value has a determinate limit, if all the ele- 
ments are finite. 

Since 

L < a n < M, 

where L and M are the least and greatest elements respectively, the mean value 
must remain finite. Also, 

I ** p 



n +P n +i n[n-\-p) i 



16 INTRODUCTION TO THE CALCULUS. [Sec. I. 

But 

/ ^ |^| ** G P G ■ 



G being an assigned number, than which no element can be greater in absolute 
value. Whatever be the assigned integer p, we can always assign an integer n 
that will make a n +f — oc n less than any assigned number e. The mean value 
therefore converges to a determinate limit. The value of this limit depends on the 
law by which the sequence is formed. 



12. Find the limit of 



K)" 



when z becomes infinite in any way whatever. 
Divide both numerator and denominator in 



X m — 



where m is a positive integer, by x m — i. Whence results 

l-fi'»-f . . . +\x m ) 



.<JV~ 



= 1 + 



l _|_ xm _j_ . . . ^x^J x~ l + x» l -f . . . +x~n 

(i). If x= I -1 , then each of the m terms in the denominator of the frac- 

/ m -\- i 

tion on the right is less than I. 

m+i 

x m — I ,i m+l 

.-. >I-h— =r — — • 

x — I m m 

Hence *^_ x x 

x nt - i > - 

m 



(. + 4rr>K)" 



Therefore, the value of the expression continually increases with m, and is 
always greater than 2, by Ex. I. 

(2). If x '— i , each of the m terms in the denominator of the same 

m 4- i 



m 4- i 



fraction is 


greater 


than i. 


i 


i 


m + I 
X~~™~ 
— X 


< I 


+ 


w 


Hence 
or 










I — X 

m + 
.* m 


f» + 

> 


< 


m* 



Art. 15.] EXERCISES. 17 

/ i \ * + 1 / 1 \ >» 

Therefore the expression continually diminishes as the positive integer m in- 
creases. 

(3). Whatever be the positive number x, we have 
X2 > x? — 1. 

... x > x + ' 



Hence (, - I) % (, + I)' 



whatever positive value jr may have. 
(4). Also, 






if we put a: = y -f- !• 

These results (1), . . . , (4), show that 

I \z 



K* 



continually increases as z increases by positive integers, and continually de- 
creases as z decreases by negative integers, and that the latter set of numbers is 
always greater than the former, by (3). Also, these ascending and descending 
sequences have a common limit,* by (4). 

The value of this limit lies somewhere between 

(1 -f 1/6)6 = 2.521 ... and (1 — 1/6)- 6 = 2.985 . . . 

We represent it, conventionally, by the symbol <?. More accurately computed, 
its value is 

e = 2.7182818285 . . . 

A more convenient method of computing e will be given later. It only remains 
now to show that the limit is the same, whether z increases by rational or irrational 
values, or continuously. 

If z is any positive number, rational or irrational, we can always find two con- 
secutive integers m and m -j- 1, such that 

m < z < m -J- 1, 

\m + 1 



and 



K^.rK^r<KH'+;rK)- 



This shows that when m = 00 , then z = 00 , and 

1 



/( 



1 +- 

z 



*Put (1 — m-*)- m = a m , [1 -f (m — i)-i]»*-i — b m . Then assigning to 
m the values 1, 2, 3, . . . . we have two sequences of positive numbers. The 
sequence a m always diminishes, the sequence b m always increases. The difference 
am — b m is a positive number converging to o when n = 00 . The two sequences 
therefore define a common limit e. 



1 8 INTRODUCTION TO THE CALCULUS. [Sec. 1. 

The result in (4) shows this is true whether z be positive or negative.* This 
limit is the most important one in analysis. 

13. Show that £ (1 + *)*"= e. 

M = )o 

14. Show that 

£ logafl +-) = £ loga(l + ^=log a ^, 

and is 1, if a = e. Use Ex. 6. 

15. Show that £ ( 1 -j- ?- j = £ (1 -f- xy)T == 0, 

* = oo\ x ) *(=)o 

16. Show that £ (a* — \)jx — log e a. 

x( = )o 
Hint. Put a* = I + z. 

17. If/// is a positive integer, show that 



£ 



— ma.™—*. 



18. Show that Ex. 17 also holds true when m is a negative integer, also if m 
is any positive or negative rational number. J_ _L 

Hint. Put/// — p/q. Divide the numerator and denominator by x* — a*?, 
to obtain the quotient in determinate form for evaluation. 

.„ 01 ■■, , r sm x — sin a TT _, , _ 

19. Show that £ ■ = cos a. Use Ex. 6, § 12. 

#( = )a ■*" — a 

20. Let/,, represent some particular one of the digits o, 1, . . . , 9, for a par- 
ticular value of r. Show that the periodic decimal 

a-p x . . . />/// + r . . . pi + m p L + z . , . . pi + m . . . 
has for its limit the rational number 

' io'(io»« — i) 

where ^/= 0.^ .. .pi, and JV = pipi + , . . . pi + M , and pi + r = pi + gm + r , 
q being any integer, and r any integer less than or equal to q. 

*The evaluation here given is a modification of one due to Fort, Zeitschrift fur 
Mathematik, vii, p. 46 (1862). See also Chrystal's Algebra, Part II, p. 77. 



SECTION II. 

ON THE FUNCTION OF A VARIABLE. 

16. Definition. — When two variables x and y are so related that 
corresponding to each value of one there is a value of the other they 
are said to he functions of each other. 

If we fix the attention on y as the function, then x is called the 
variable ; if on x as the function, then y is called the variable. 

Such functions as x and y defined above are not amenable to 
mathematical analysis until the law of connectivity between them can 
be expressed in mathematical language. 

Classification of Functions. 

Functions are classed as explicit or implicit functions according as 
the law of connectivity between the function and the variable is direct, 
explicit, or indirect, implied, implicit. 

17. Explicit Functions. — The simplest form of a function of a 
variable x is any mathematical expression containing x. Such a 
function is called an explicit function of x, because it is expressed 
explicitly in terms of the variable. 

Our attention will be confined in Book I principally to explicit 
functions of one variable. 

The three standard or elementary functions, 

x a , sin x, \og a x, 

and their inverse functions, 

xr a , sin _I x, a x , 

represent the three fundamental classes of functions called algebraic, 
circular, and logarithmic or exponential. All the elementary explicit 
functions of analysis are formed by combining these standard functions 
by repetitions of the three fundamental laws of algebra, 

Addition, Multiplication, Involution, 

and their inverses, 

Subtraction, Division; Evolution. 

Explicit functions are classified as algebraic or transcendental accord- 
ing as the number of operations (including only 

addition, multiplication. involution, 

subtraction, division, evolution, 

by which the function is constructed from the variable), is finite or 
infinite. 

19 



20 INTRODUCTION TO THE CALCULUS. [Sec. II. 

1 8. The Explicit Rational Functions. 

I. The Explicit Integral Rational Function. 
The function of the variable x t 

a + a x x +"*** + • • • +a n x n , 
where the numbers a , . . . , a H are independent of x, and n is a finite 
integer, is called an explicit integral rational function of x, or briefly a 
polynomial in x. 

This is the familiar function which is the subject of inquiry in the 
Theory of Equations. Its place and properties in the system of func- 
tions correspond in many respects to the place and properties of the 
integer in the system of numbers. It can advantageously be expressed 
by the compact symbolism 

n 

^5 a r x r , 

r—o 

meaning the sum of terms of type a r x r from r=otor=». 

II. The Explicit Rational Function. 

The quotient of two explicit integral rational functions of a vari- 
able x, 

a o + <yv -f . . . + a n x n 

UV'+ • • • +4-*"' 

is called an explicit rational function of x, or simply a rational function 
of x. 

Its place in the system of functions corresponds to that of the 
rational or fractional number in the number system. 

III. The Explicit Irrational Algebraic Function. 

Any expression involving a variable x, or an integral or rational 
function of x, in which evolution a finite number of times (fractional 
exponents) is the only irrational part of the construction, is said to be 
an explicit irrational algebraic function of-r. 

Such a function in the function system corresponds to those irra- 
tional numbers in the number system called surds. 
For example, 

\'a* — x 2 , a + bxi, Vi-\-x/ \ i — x, 

arc irrational algebraic functions. 

19. Explicit Transcendental Functions. — Any expression which 
is constructed by an infinite (and cannot be constructed by a finite) 
number of algebraic operations on a variable x is said to be an ex- 
plicit transcendental function of .v. 

Examples of such functions are sin x, e x , log x, tan - ■ x, etc., which can only be 
constructed from j by an infinite number of operations, such as infinite series or 
products, or continued fractions. 

20. Implicit Functions. — Whenever we have any equation involv- 
ing two variables, x andj', this equation is an expression of the law of 



Art. 2i.] ON THE FUNCTION OF A VARIABLE. 21 

connectivity between the two variables and defines one of them as a 
function of the other. The functional relation is implied by the 
equation and is not explicit until the equation is solved with respect 
to one or the other of the variables. 

For example, the equation 

ax 2 + by* - c = o 

defines x as a function of y, and. just as much so, y as a function of x. These 
functions can be expressed explicitly by solving for x and y. Thus we have 



\c - by* \ c 



- ax* 

~b~ 



or x and;' are expressed as explicit irrational algebraic functions of each other. 

In general, any algebraic polynomial in two variables x andj' 
when equated to zero defines^ as an algebraic function of x, and x 
as an algebraic function of y. The explicit algebraic functions of 
§ 18 are but particular cases of this more generally defined algebraic 
function. 

ai. Conventional Symbolism for Functions. — We frequently 
have to deal with a class of functions having a common property 
or common properties, and with functions of complicated form, 
which makes it convenient to adopt abbreviated symbols for func- 
tions. Thus, we frequently represent a function of the variable x 
by the symbol f(x), or F(x), <fi(x), tp(x), etc., when it is necessary 
or advisable to indicate the variable and the function in one com- 
pact symbol. When the variable is clearly understood, the paren- 
thesis and the variable are frequently omitted and the function symbol 
written/", F, or t/:, etc. 

We frequently employ the symbols y, z, u, v, etc., as functions of jr. 

In like manner we write a function of two variables x, y as (p(x, y) 
or f(x,y), etc., meaning a mathematical expression containing x and 
y. The equation 

<p(x,y) = o 

implies, as said before, a functional relation between x and y, and 
defines_yas an implicit function of x, or .a; as an implicit function ofy. 

ltf(x) is a function of x, and if a is any particular assigned value 
of x, we write f(a) as the value of the function when x = a, or, as we 
say, the value of/(.%-) at a. 

For the present, when we use the word function we mean an 
explicit function of one variable. 

A function, /[x) 9 is said to be uniform or one-valued at a when the 
function has one determinate value at a. 
For example, 

ax 1 -\- bx A- c , e x y sin x, 

are one- valued functions for any value of x. 

\{ f{pc) has two, three, etc., distinct values corresponding to a 



22 INTRODUCTION TO THE CALCULUS. [Sec It 

value of the variable, it is said to be a two- , three- valued, etc., func- 
tion. 

For example, ax*, \/a 2 — x 2 , are two- valued functions of x. 

Frequently a function does not exist (in real values or finite values) 
for certain values of the variable. Then, it is necessary to define the 
interval of the variable in which the function does exist and in which 
the investigation is confined. 

For example, the function .\/a 2 — x 2 exists as a real function only in the inter- 
val ( — a, -\- a); the function represented by the series 

exists as a determinate finite function only in the interval ) — I, -f- i(. 

22. Continuity of a Function. 

Definition : f(x) is said to be a continuous function of .rat x = a y 
when_/*(.v) converges to f(a) as a limit, at the same time that x con- 
verges continuously to a as a limit. 

The definition and condition of continuity of f(x) at a are com- 
pactly expressed in symbols' by 

£A*) =/U»- 

The function f(x) is said to be continuous in an interval (a, ft) 
when it is continuous for all values of x in (a, /?). 

The definition of continuity of f(x) at x asserts that whatever 
absolute number 6 is assigned, we can always assign a corresponding 
absolute number h such that for all values of x 1 satisfying the inequality 

we have 

\A*d-A*)\ <*■ 

Since, by definition, the limit of/[x^) is/(x), we can make and 
keep 

\A*d-A*)\ 

less than any assigned absolute number d for all values of/"^) sub- 
sequent to an assigned value f(x'). 

If x -f- h is the value of the variable corresponding tof(x'), then 
all the values of the function corresponding to the values of the vari- 
able in (x, x 4- h) satisfy the inequality above. 

An important corollary to the above and a principle which will 
constantly be employed later is: If/"(tf), the value o\ f(x) at a, is 
different from o and is finite, then we can always assign a finite num- 
ber h such that for all values of x in the interval (a — h, a -J- h) the 
function J\x) has the same sign as f{a). 

The above definition shows that a continuous function must change 
its value gradually as the variable changes gradually, and that the dif- 
ference of the values of the function 

/K) -A*,), 

must be arbitrarily small in absolute value when the difference of the 
corresponding values of the variable, x l — x 2 , is arbitrarily small. 



Art. 23.] ON THE FUNCTION OF A VARIABLE. 23 

It also shows that a function cannot be infinite at a value of the 
variable for which the function is continuous, and vice versa. 

In symbols, when f(x) is continuous at a, we must have simulta- 
neously 

£{x -a) = o and £[/(x) -/(a)] = o. 

The definition and condition of continuity at a can be expressed in 
the compact symbol 

£/{x) =/(a). 

23. Fundamental Theorem of Continuity. — If f{x) is a uniform 
(§21) and continuous function of x in an interval (a, b), then what- 
ever number iY T be assigned between the numbers /{a) and/*(^), there 
is a value B, of x in (a, b) such that at B, we have 

/(£) = N. 

The proof of this theorem falls under two heads. 

I. If a function f(x) is one-valued and continuous throughout an 
interval (a, b), and_/(tf) and/" (b) have contrary signs, then there is a 
number B, in (a, b), at which we have 

/(*) = o. 

Suppose /{a) is negative and/*(3) positive. Then/(.*) cannot be 
o or -f- arbitrarily near to x = a, nor canf(x) be o or — arbitrarily 
near to x = b, by definition of continuity ofy r (^) at a and b. 

Let b — a = h. Divide this interval into 10 equal parts by the 
numbers 

a, a + T \/i, . . . , a -f T %k, b. 

Either f(x) is o for x equal to one of these numbers, in which 
case the theorem is proved, or it is not. In the latter case let a be 
the last of these numbers, proceeding from the left, at whichy(jr) is 
negative, and b the first at which it is positive. 

Proceed in exactly the same way, subdividing the interval (a , 5) 
into 10 equal parts. Then \{/{x) is not o at one of the new division 
numbers, let a 2 be the last at which it is negative and b 2 the first at 
which it is positive. 

Continuing this process n times, we find that either/* (x) is o at 
one of the interpolated numbers, or \haXf(a n ) is negative and f{by) is 
positive (see § 15, VII, 3), and 

n n 

a n = a + k Y^- , b n = a + h V ' *! + A, 

Z> io 1 - Z> io io 

where each p r (r = 1, 2, . . . , n) represents some one of the digits 
o, 1, . . . , 9. If f(x) is o for some one of the interpolated 
numbers obtained by continually subdividing (a, b), the theorem is 
proved; if not, then the. two numbers a n and b n , the former always 



24 INTRODUCTION TO THE CALCULUS. [Sec. II. 

increasing, the latter always diminishing, converge to the. common 
limit 

Meanwhile /(a,) and /(b n ) converge to the common limit f(Z), 
by the definition of continuity. The first of these f(a n ) is always 
negative, the second /(b H ) is always positive. Also, since b n — a n = 
h/io n , we must have 

£\/{K) -/(«.)} = £\ |/(*.)l + l/(«.)l I. 

= 2\f(S)\. 

But this limit is o, by definition of continuity. 

.-. f(£) = o. 

In like manner we prove the theorem when /(a) is positive and 
f (b) is negative. 

II. The general theorem now follows immediately. For, what- 
ever be the numbers / (a) and /(b), if N lies between them, then 

/(*) - n 

must have contrary signs wheme = a. x = b. Therefore, by I, there 
must be a number B, in (a, b) at which 

f{Z)-N=o. 

The important fact demonstrated by this theorem is this : If a 
function f (pc) is uniform and continuous in an interval (a, b) of the 
variable, then as the variable x varies continuously through the inter- 
val (a, b), the function must vary continuously through the interval 
determined by the numbers /*(#) and/*(£). That is, the function f(x) 
must pass through every number between /(a) and /(b) at least 
once. 

24. General Theorems. — The following general theorems result 
immediately from the theorems on the limit, § 15, and the definition of 
a continuous function. 

I. The sum of a finite number of continuous functions is a con- 
tinuous function throughout any common interval of continuity of 
these functions. 

If/j(jr), / 2 (x), . . . ,/ n (x), are continuous at x, then 

*(*) =/,(*) + ■ • • +/.(*) 

is a continuous function at x. For we have 

£<P(x') = £[/ l (.v')+ . . . +/„(*')]. 

= £/M-') + • • • ;£/.(•*')■ 
=/iG6*0 + • • • /«(£*")> 

= /,(•') + • • • +/«(*) = 0(*). 



Art. 24. j ON THE FUNCTION OF A VARIABLE. 25 

II. The product of a finite number of continuous functions is a 
continuous function in any common interval of continuity of these 
iunctions. 

if 0W-/W •/,(*) • • •/.(*), 

then f*(*)=£\SJP) ■ ■ •/.(*')]. 

= £AW) ■ ■ ■ £/.(*'), 
=£(£*') ■ ■ ■ A(£x'), 
=/iW • • • /«(*) = «(*)• 

Corollary. Any finite integral power of a continuous function is a 
continuous function in the same interval of continuity. 

III. The quotient of two continuous functions is a continuous 
function in their common interval of continuity, except at the values 
of the variable for which the denominator is zero. 

If f(x) = (p(x)/ip(x), then we can consider/^) as the product of 
<f>(x) and i/(J:(x). The theorem is then true by trie reasoning of the 
preceding theorem. 

Otherwise, 

r fix'\- r* {x,) - £ ^ x,) 

provided ip(x) ^ o. If tp(x) = o and cp(x) =£ o, then/"(jr) = 00 and 
is not continuous at x. If ip{x) = o and also <fi{x) = o, then f(x) 
may or may not have a determinate value as the limit of J\x f ), a case 
which we shall investigate later. 

IV. It has been shown, Exercises, Sec. I, Ex. 5, that 

r a f{.*) — a im — a f { £*\ 

when a is positive. Therefore a f{x) is continuous when f(x) is con- 
tinuous. 

V. In like manner, Ex. 6, Exercises, Sec. I, 

£ log.,/0) = \og a £f{x) = \og a /(£x), 
f(x) being positive. Therefore, \og a f(x) is continuous. 

VI. Again, it f\x) and <fi{x) are continuous and f(x) is positive, 
we have 

y = [Ax)Y* x) - 

• •■ logy =</>{x) log f(x), 

£**> = £&(*) kg A*)1, 

= £4>(x)-£\og/(x). 

... log£y=<M£x)logA£x), 
= ^g [A£x)1# ix) . 
... £[/{x)Y^ = [_/(£ x )Y M , 



26 



INTRODUCTION TO THE CALCULUS. 



[Sec. II. 



and the function y is continuous when (p(x) is continuous and f(x) is 
continuous and positive. 

Special Theorems. 

Since y = x is a continuous function of x, the product a r x r , where 
a r is independent of x, and r is any finite integer, is continuous for all 
finite values of x. Also the sum of any finite number of terms of this 
type is continuous. Therefore the algebraic polynomial in x is a 
continuous function for all finite values of x. 

By the theorem for the quotient, it follows that the algebraic frac- 
tion or rational function is continuous everywhere, except at the roots 
of the denominator. 

By Trigonometry, since 

sin x' = sin x -f- 2 cos ±(x' -f- x) sin %(x' — x), 
and jQ sin \{x' — x) = o, when x'(z=)x f 

we have £ sin x' — sin x = sin £x. 

Therefore sin x is everywhere continuous. 

In like manner we show that cos x is everywhere continuous, and 
by § 24, III, tan x, cot x, sec x and esc x are continuous functions 
everywhere except at the roots of their denominators, cos x, sin x. 

The continuity of any algebraic function of 

x a , a x , sin x, log x, 

can now be easily determined. 

25. Geometrical Illustration of Functions. — If we adopt the method of rep- 
resenting the variable, in §§8, 11, by points on a straight line, such as Ox, then at 
any point M on Ox corresponding to x = a we can represent the corresponding 



j ; 



M* 



M 7 3L 3L O 



P 
5? 



3!i M 2 



Pi 

Fig. 4. 



4 



M* M M L 



-x 



value of a uniform function /(x) by a point Pina plane xOy. The point Pis con- 
structed by laying off a perpendicular MP to Ox, such that the distance MP is 
equal to the number f{a), and is measured upward if f{a) is positive, and down- 
ward if /(a) is negative. 

For each and every value of x for which f{x) is a defined function, such as <7j , 
o 2 , . . . , we can construct corresponding points P x , P 2 , . . . , representing /(a,), 
/(fl 2 ), . . . 

This is the familiar method of Analytical Geometry, invented by Descartes. 
If we put y = f(x), then Oy perpendicular to Ox can be called the axis of the func- 
tion, corresponding to Ox. the axis of the variable; and x, y are the Cartesian coor- 
dinates of the point P representing the functional formf(x). 



Art. 25.] 



OX TIIK FUNCTION OF A VARIABLE. 



27 



If the function /(.v) is continuous in any interval (a., or 6 ), then corresponding to 
any point J/ i; in M^M & there is a point P 6 in the plane xOy, at a finite distance 
from Ox, representing the function. Moreover, any two such points / y , P" cor- 
responding to J/'. M" can be brought as near together as we choose by bringing 
M' and M " sufficiently near together. Can we say that the assemblage of all the 
points, P. representing a continuous function in a given interval (a, /;) of .v, is a 
line ? 

To answer this question it is necessary to consider the question : What con- 
stitutes a line, or in general a curve? 

Geometrically speaking, the older definitions, now antiquated, required a line 
to have in the first place a determinate length corresponding to any two arbitrarily 
chosen points on the line, and also to have direction at any point. This requires 
a definition of direction and of length, concepts themselves abstruse. The old 
definition, (i a line has length without breadth or thickness," is now taken to mean 
that a line is simply extension in one dimension. 

In order that the assemblage of points in the plane xOy representing a con- 
tinuous function f(x) can be defined as a line, this assemblage must have some 
analytical property at each point that will define a determinate direction, and cor- 
responding to any two points some analytical property that will define a determi- 
nate length. These properties must be inherent in the function f(x) of which the 
assemblage of points is the geometric picture. 

To define the first of these properties, i.e., a determinate direction, is the prov- 
ince of the Differential Calculus ; the second, which gives meaning to a definite 
length, is furnished by the Integral Calculus. 

At our present stage of knowledge, then, we cannot say that the assemblage of 
points which represents a continuous function is a line. But it will be demonstrated 
in what follows that such continuous functions as those with which we shall be con- 
cerned can be represented by curves, and we shall in the course of our work 
develop an analytical definition of a line, and find means of measuring its direc- 
tion, length, and curvature, and many other properties that are unattainable save 
through the Calculus. 

In order to take advantage of the intuitive suggestiveness of geometrical pictures 
as illustrations of the text, we shall assume for the present that the assemblage of 
points P x , . . . , P n representing values of a continuous function in the interval 
M Y M n , has the following properties : 




Join the consecutive points by straight lines. Consider the broken polygonal 
line P x P 2 . . . P n . Then, if M y and M n correspond to two fixed values a, b of x, 
and we increase the number of points, Af, between M x and M n indefinitely, in such 
a manner that the distance between any two consecutive points M r and M r + r con- 
verges to zero, we shall have : 

First. The distance between the corresponding points P r and P r + t converges 
to o. For, P r P r + 1 is the hypothenuse of a right-angled triangle, P r NP r ,, 
whose sides P r A^and IS/P r + 1 =M r M r + x converge to o together, when M r ( = )M r + u 
since the function f[x) is continuous.* 



* The point N, not shown in the figure, is the point 
through P r + x parallel to Ox cuts M r P r - 



which a straight line 



28 INTRODUCTION TO THE CALCULUS. [Sec. II. 

Second. We assume that the angle P r ^ l P r P r ^. x between any pair of con- 
secutive sides of the polygonal line, such as P r _ v P r and P r P r+l , converges to two 
right angles as a limit. 

Third. We assume that the sum of the lengths of the sides of this polygonal 
line P x P n converges to a determinate limit length. 

The first consideration secures continuity, the second determinate direction, and 
the third determinate length. 

The three together constitute the necessary conditions that the assemblage of 
points shall be a curve. 

The analytical equivalents of the second and third conditions will be developed 
later. That for the first has already been established in the definition of a con- 
tinuous function. 



Art. 25.] EXERCISES. 



EXERCISES. 



29 



1. If f[x) — 2X 3 — x 2 — I2x -f 1, show that the function has a root in 
each of the intervals (o. ij, (2, 3), (— 3, — 2). 

2. If 0(x) = (x - i)/(x -f 1). show that 

(pa) — 0(b) _ a — b 
1 + 0(a)0(b)~ 1 +ab' 

3. If #(/) = et -j- <?- *, show that 

1K30 = WO] 1 ~ 3#), 
^ + /) X #* -J) = ^(2x) + rp{2y). 
1 x 

4. If F[x) = log , show that 

5. What functions satisfy the functional equations 

f(x+y)=f(x).f(y), 
0(x) -j- 0(/) = 0(xy), 
4>(x) - tf>{y) = TP\x/y), 

F(x-y) = F(x)/F(y). 

6. If f{x) = #x 2 — 3x -f- <r, write /(sin x). 

7. If / = x 2 -f- x — 5, write xasa function of y. 

1 

8. Show that e x is discontinuous at x = o. Examine the behavior of this func- 
tion as x increases through o. 



9. If y — log (x -f \/x 2 -f 1), show that 

x = |(^ _ <? -*). 

This last function is called the hyperbolic sine of y and is written 
sinh y EE ^(^ — e — >). 



10. If/ = log (x -)- fx 2 — 1), x is called the hyperbolic cosine of y and 
written cosh y. Find this as a function of y. 

11. Show that <?:f = sinh y -\- cosh/. 

12. Let x be any assigned real number. Consider the function 

where n takes only positive integral values. Show that Fin) has the limit o when 
11 = 00 , whatever may be the finite value of x. 



13. Show that 



aja-D.. (a -r+i) 



in which a and x are assigned real numbers, has the limit O when r = 00 , pro- 
vided |x| < 1. What is the value of/(oo ) when \x\ > I ? 

14. Investigate 

x« 

W =00 

for |x| £ 1. 



/x n 



30 INTRODUCTION TO THE CALCULUS. [Sec. II. 

15. The identity 



ab 



ta + b^ la— bV' 



shows that the geometrical mean, \/<zb, of two unequal numbers lies between them 
and is less than their arithmetical mean \{a -f- b). 

Finding the square root of any absolute number /j can be reduced to finding 
the square root of a number between I and ioo. For, we can always assign an 
integer n such that \o 2n fi = a, where I < a < ioo; n being -f- or — according 
as fi is less or greater than I. Then 

\/JS = 10 - n |/a. 

Consider any given number between I and ioo. Choose x 1 from one of the 
integers 2, . . . , io, such that 

(x, — i) 2 < a < x*. 

Then — < \/a < x. , 

x • 



a ,- 1/ , a\ 



Show that if this construction be continued to x m , then 

x m - Va < ^-^ , 

and therefore the sequence of numbers x x , x 2 , ... defines the square root of a, 
and 

£ x m = \/a. 

m = co 

16. Apply 15 to show that 4/5, to six decimal places, is 

2207 
** = W = 2 ' 2 3 6o68 9- 

17. Show that the cubic function of x, 

a -x, h , g I =/(x) 

h , b — x, f 

S , / , c-x\ 
always has three real roots. 

Expanding with respect to the first row, 

fix) = (« - x)[(b - x)(c - x) -/*] - [#(«: - *) - 2fgk+g*{b - x)\ 

Let />, </, of which / is not less than q, be the two roots of the quadratic 
function 

(/, _ X )( C _ x ) _ p = #2 _ (£ _[> <-) x 4 *; - /*. 

Then / -f- $r = b -f- r, and /</ = be — /-. 

Therefore neither / nor q can be between b and c or equal to b or *", and p is 
greater and ^ is less than either b or c. But 

/(+ 00 ) = - 00 , 

AP) = + \h V p-c + g \'P - b-\\ 

f{q) = -[bVc-q-gVb-q?, 

./(-oo) = +oo. 

Hence, by §23, I,/(x) vanishes between -J- 00 and/, between / and q, also 
between q and — do , and the three roots are real. This exercise will be needed 
in subsequent work. 



Art. 25.] EXERCISES. 31 

18. Determine the condition that the function 

ax* -\- 2bx -\- c 

shall retain its sign unchanged for all values of the variable x. 
The function can be written 

_ {OX -f J)* 4- (fltf — ^ 2 ) 

— a 

In order that this shall retain its sign unchanged for all values of x, it is 
necessary and sufficient that ac — U 1 shall be positive. This condition being satis- 
fied, the function has the same sign as a for all values of x. 

19. Determine the condition that the function 

ax 2 + ihxy + by* 

shall retain its sign unchanged for all values of the variables x and y. 
By completing the square, the function can be written 

(ax 4- hyf + y\ab - h 2 ) 

a 

which, when ab — h 2 is positive, has the same sign as a for all values of x &ndy. 

20. Determine the condition that the function 

ax' 2 -)- by 2 -j- cz 2 -\- 2fyz ~\- 2gxz -\- 2hxy 

shall keep its sign unchanged for all values x, y, z. 

By completing the square, the function can be written 

I- J {ax + gz + hyf + {ab - h 2 )y 2 + 2{fa - kg )y z + {ac - g 2 )z 2 j. . 

The function will keep its sign unchanged and have the same sign as a what- 
ever be the values of x, y, z, provided the quadratic function 

(ab - h 2 )y 2 4- 2(fa - hg)yz 4- (ac - g 2 )z 2 

is always positive. This will be the case, by Ex. 19, when 

ab - Ji 1 and (ab - h 2 )(ac - g 2 ) - (fa - kg) 2 

are both positive. Or, what is the same thing, ab — A i and 

a(abc 4- 2fgh — a/ 2 — bg 2 — ch 2 ) = a\ahg 

\hbf 
\gfc 
must be positive. 

Exercises 18, 19, 20 will be drawn on in the sequel. 



33 



BOOK I. 
FUNCTIONS OF ONE VARIABLE 



PART I. 

PRINCIPLES OF THE DIFFERENTIAL CALCULUS. 

CHAPTER I. 
ON THE DERIVATIVE OF A FUNCTION 

26. The Difference of the Variable. — The difference of a variable 
x is a technical term, which means the result obtained by subtract- 
ing a particular value of the variable, say x, from an arbitrarily 
assigned value of the variable, say x v 

Or, in symbols, 

x 1 - x. 

We use the characteristic letter A to represent the symbol of this 
operation, and write 

Ax = x x — x. 

This difference, Ax, is of course positive or negative according 
as x x is greater or less than x. 

We sometimes for convenience write 

Ax = x x — x = h, 
so that 

x x = x -\- h, 

and call h the increment of the variable x. 

27. The Difference of the Function. — The difference of the func- 
tion is a corresponding technical term, which means the result 
obtained by subtracting the value of a function at a particular value 
of the variable, say x, from the value of the function at an arbitrarily 
chosen value of the variable, x v In symbols 

/(■*,) -■/(■*) 

is the difference of the function f{x) at x. 

As in the case of the difference of the variable, we use the letter 
A as the symbol of this operation, and write 

Af(x) =/(^) -f{x). 

35 



36 

28. The Difference-Quotient of a Function. 

A difference of a function and a difference of the variable are 
said to " correspond " when the same values of the variable occur 
in the same way in these differences. 

Definition.— The quotient obtained by dividing a difference of 
the function by the corresponding difference of the variable is called 
the difference-quotient of the function at the particular value of the 
variable. 

Thus, in symbols, 

4/1*) -A*d -/(•*) 



Ax 



<h 



is the difference-quotient of the function f{x) at x. 

For an assigned particular value x, the number q x depends on 
the value assigned to the arbitrary number x v 

29. The Derivative of a Function. 

Definition. — Whenever the function /{x) is such that when we 
assign to the arbitrary value of the variable successive arbitrarily 
chosen values • 

"^1 ' ^2 » * * * ' ^ n » * * • 
in such a manner that this sequence converges to the particular value 
x as a limit, and the corresponding sequence of difference-quotients, 

A*\) -A*) _ A*J -A*) _ a A*n) -/(•*) _ a 

x x -x -* 1 ' x 2 -x -^"•" Xn - X -?»,-.• 

has a determinate number as a limit, this limit is called the deriva- 
tive of the function f{x) at x. 

In other words, the function_/(.r) is said to be differ entiable at x 
when the difference-quotient 

Af(x) _ f(x')-f{x) 

Ax x' — x 

converges to a determinate limit as x' converges to x as a limit in 
any arbitrary manner whatever. 
In symbols, 

PA *') -/(*) 

is called the derivative oif(x) at x. 

This derivative is, in general, a function of .v, and we shall 
represent it, after Lagrange, by the symbol f'(x)> a convenient and 
characteristic symbolism because it shows the association of the 
derivative f\x) with the primitive function f{x) from which it has 
been derived. 



Art. 30.] ON THE DERIVATIVE OF A FUNCTION. 37 

We shall also use sometimes another symbolism, to represent the 
operation by which this limit is derived, instead of the cumbersome 
one employed above representing the limit of the difference-quotient. 

We use the characteristic letter D as a symbol to represent the 
operation gone through of dividing the difference of the function by 
the corresponding difference of the variable, and determining the 
limit of this difference-quotient when the arbitrary value of the 
variable converges to the particular value of the variable as a limit. 

In compact symbols, we write 

x'( = )x 

But we have already agreed that this limit, the derivative, shall 
be represented by f\x). Hence we have the equivalent symbolism 

D/(x) =/'(x). 

Or, the operation D performed on the function f(x) results in 
the derivative f'(x). 

This operation is called differentiatio?i. 

30. Observations on the Derivative. — We observe that in order 
that a function f{x) may be differentiate (have a derivative), it must 
be continuous. For, unless we have 

x'( = )x x'( = )x 

as is required by the definition of a continuous function, then, since 
we do have 

£{ x ' - *) = °, 

xf( = )x 

the value of the corresponding difference-quotient would be 00 , or 
no limit exists. 

Hence the Differential Calculus deals directly with none but 
continuous functions. 

The converse of the above statement is not true, i.e., a function 
that is uniform and continuous is not always different! able. There 
exist functions that are uniform and continuous and yet the limit of 
the difference-quotient is completely indeterminate for all values of 
the variable in certain finite intervals. * We shall not have occasion 
to meet any of these highly transcendental functions in this book, 
and the functions with which we deal will, in general, be differentia- 
ble. Only for isolated values of the variable will the derivatives of 
these functions be found indeterminate. Such values are singular 
values and receive treatment in their appropriate places. 

The evaluation of the derivative of a function falls under the 
case specially excepted in § 15, V. Here, the limit of the numerator 

* See Appendix, note I. 



38 FRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. I. 

(the difference of the function, 4/)> an< ^ tne limit of the denomi- 
nator, Ax, are each o. 

The quotient of the limits o/o is always indeterminate. 

We are not concerned, in evaluating the derivative, with the 
quotient of the limits, but only with the limit of the quotient. 
We are not concerned or interested in the differetice-ratio but with 
the difference-quotient. 

This is a variable number which does or does not have a limit 
according as the function is or is not differentiate at the particular 
value of the variable considered. 

The derivative of any con stant is necessarily o by the definition. 
For, the quotient of differences is constantly o and remains o for 
Ax( — )o. 



EXAMPLES. 



1. Differentiate the function x 2 . 
We have the difference-quotient, 



x ' — x 



= X -\- X. 



X — X 

The limit of this number when x'(=)x is 2x. 

. •. Dx 2 = 2X. 

2. Differentiate the function x". 



We have 



x' h - x l _ I 

x' — x y 2 -\- x* 



the limit oi which is x */2 when x'(=.)x. 
'... Zte* 

3. If f(x) ~ s i n x \ show that D sin x = cos x. 

We have, by Trigonometry, 

x' — sin x = 2 cos l(x' -j- x) sin ^(x' — x). 

sin x' — sin x sin l(x' — x) 

; = cos Ux 1 -4- x) , -\ — '. 

x' -x * v ^ ' i(^ _ jr) 



sin 



■^r =I ' 



/ sin *(x' 



*'( = )* 

. \ f'{x) = cos jr. 

4. Show that the derivative of any constant is zero. 

If A is any constant, it keeps its value unchanged whatever be the value of jr. 
Therefore the difference-quotient is 

A - A 
. = o 

x x — X 
for all values of x x =£ jt and when jr x ( = )x. Consequently DA = o. 

5. Show that the derivative of the product of a constant and any function of x 
is equal to the product of the constant and the derivative of the function. 



Art. 31.] 



ON THE DERIVATIVE OF A FUNCTION. 



39 



Let a be constant and y a function of x. Let y take the value y y when X takes 
the value x v The difference-quotient of «y is 

<iy x — ay _ \\ — y 



the limit of which is aDv. 



Da\ 



aDy. 



31. Geometrical Picture. — We have seen that a differentiable 
function is necessarily continuous. We shall now see that the as- 
semblage of points taken to represent it possesses the characteristic 
property of a determinate direction at each point and can be considered 
as a curve. 




In the figure, MP, P' represent /(or), f{x'), then 
Ax — x' — x — MM', 
4f(x)=f(x')-f(x)=NP', 
A/(x) NP' 



Ax 



PN 



= tan 6', 



where 6' = /_ NPP' is the angle which the secant PP' makes with Ox. 
By the definition of a tangent to a curve, the limiting position of the 
secant PP' as the point P' moves along the curve and converges to P 
as a limit is the tangent PT to the curve at P. At the same time 6' 
converges to 6 as a limit, 6 being the angle which the tangent PT 
makes with Ox. But tan 6 is the limit of tan 6' ', and is therefore the 
limit of the difference-quotient, or is the derivative of f(x) at x. 
Therefore we have 

D/{x) =/'{x) = tan d. 

Hence the derivative of a function is represented by the slope of 
the tangent to the curve which represents the function. The direc- 
tion of a curve at any point on it is the direction of the tangent there, 
and the slope or declivity of the curve is that of its tangent at the 
point. 



I , 



40 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. L 

32. Ab Initio Differentiation. — The process of differentiating a 
given function directly from the definition by evaluating the limit of 
the difference-quotient is, in general, a complicated and tedious pro- 
cess. We shall in the next chapter deduce certain rules of differentia- 
tion by which, when once we have differentiated log x and sin x by 
the ab initio process, we can write down directly the derivatives of all 
the elementary functions in terms of these derivatives and those 
which follow. Meanwhile, in order not to lose sight of the ab initio 
process and the rationale of differentiation which is at bottom always 
the evaluation of a limit, the limit of the difference-quotient, the fol- 
lowing exercises are set for solution by this method. 





EXERCISES. 




Differentiate the following functions : 




1. 3-r 2 - 6x. 




6(x - 1). 


2. 7^ - 13. 




28X 3 . 


3. (x - l)(2* -f 3). 




4-r + 1. 


4. x-*. 




— *-*. 


5. ax— 3. 




- 3«X"4. 


6. {x - a)/(x + a). 




2a(x -f- a)— 2 . 


7. *■• 




8,4 


8. (x 2 - 2)2. 




x(x 2 — 2)~K 


9. 2(X + 1)-*. 




- (x + 1)-*. 


10. A 




*HL 


11. x n . (it any finite 


integer.*) 


«x w -i. 


12. x<J. (p and q positive integers.*) 

13. cos X, 


p i-i 

— sin x. 


14. tan x. 




sec 2 x. 


15. log x. (See § 15, 


Ex. 11.) 


x— 1 . 


16. sec x. 




sec jt tan x. 


17. fl*. (Use Ex. 15, 


§150 


a* log, tf. 


18. x*. (x positive, a rational.) 


«jr a_ 1. 



* Divide numerator and denominator of the difF.-quot. by x x — x in Ex. 11, 

1 1 

and by xf — x q in Ex. 12. 



CHAPTER II. 
RULES FOR ELEMENTARY DIFFERENTIATION. 

33. As was stated in Chapter I, when we have once differentiated 
.r*, sin x, log x, by the ab i?iitio process, we can differentiate directly 
any elementary function of these functions by certain rules for 
differentiation, without recourse to the ab initio process directly.* 
These rules are themselves deduced by that process, and their appli- 
cation to differentiation is but a short method of evaluating the 
limits which we call derivatives. We shall see that the direct differ- 
entiation of only two, sin x and log x, are necessary, for X 1 can be 
differentiated by means of log x. Independent proofs, however, are 
given in each case. 

34. Derivative of log a x. — We have for the difference-quotient, 
writing x l — x = h, 

log.(* + A)-log.* = i^ i+ ^ 

writing x/h = z. When h{=)o, z = 00 . 

= \^£{ 1 +l)- §'5, Ex. 6. 

Z = 30 

* As a matter of fact, the evaluation of only one of these functions, log x, by 
the ab initio process is necessary. That is, the differentiation of all functions can 
he reduced to the evaluation of the single limit, (1 -|- i/x)*, when x = 00 , 
§ 15, Ex. 12. For, the differentiation of log x gives that of e x , and we have 

sin x — —.{e ix — e~ {x ), where ?EE+ V — 1 - We do not, however, recognize com- 
plex numbers in this book directly, which necessitates an independent differentia- 
tion of sin x, and restricts us to a geometrical definition and differentiation of that 
function. 

41 



42 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II. 



The evaluation of this limit is effected in § 15, Ex. 12, and is 
<he number 

1 -| I =«=2' 7182 



/ 



. \ D log, x = - log a e. 



In particular, if a = e, then log,, e = 1, and 
Z> log .v = -. 

According to common usage, when the base of the logarithm 
employed is e we omit writing the base and put log x for log,, x. 

35. Derivative of x a . — Let a = p/q, where p and q are positive 
integers. 

Dividing the numerator and denominator of the difference- 
quotient 

L ± 

x,t — x <? 



by x x 9 — x? , the difference quotient-becomes 

\xj) + (^7) x7 + . . . + ^7(^7) -f (jv7J 

In the numerator there are p terms each of which has the limit 



UT\ 



and in the denominator there are q terms each of which 

has the limit \xi) , when x 1 {=)x. Therefore the limit of the 
difference-quotient is 



Dx< = 



^ = l( x 7f 



tf) 



9 „±Y 9 



9. 

If a = — p/q, then the difference-quotient is 

p p P P 



p 
1 



x — x 

X, <7 X 9 l 



the limit of which for x x ( = )x is, by the above, 



2- ~Xq = — ^-A- * , 



Art. 36.] RULES FOR ELEMENTARY DIFFERENTIATION. 43 

Therefore, whatever be the rational number a, 

Dx* — (l.Y>-\ 

Rule: Multiply by the exponent and diminish the exponent by 1. 

36. Derivative of sin x, cos x. — It has been shown in Chapter I, 
§ 30, Ex. 3, that 

D sin x = cos x. 

The derivatives of all the other circular functions can and should 
be deduced in like manner. They can, however, as we shall see, all 
be deduced from that of the sin x. 

For immediate use we have, from Trigonometry, 

cos x' — cos x = — 2 sin \(x' -f- x) sin \(x' — x). 
cos x' — cos x •-,/,, x sin H x ' — x ) 

Hence, on passing to the limit, 

D cos x = — sin x. 

Rules for Differentiation. 

37. We proceed to establish rules for the derivative of the (1) sum, 
(2) product, (3) quotient, (4) inverse function, and (5) function 
of a function, in terms of the derivatives of the functions involved. 

These are the general rules for the differentiation of all functions 
with which we shall be concerned. It is necessary to know them 
perfectly, for they are the tools with which the Differential Calculus 
works. 

38. Derivative of an Algebraic Sum. 

Let y = u -f- v -f- w, 

where u, v, w are differentiate functions of x. Let the differences 
of these functions be Ay, Au, Av, Aw, respectively, corresponding 
to the difference Ax of the variable x. Then, if y, u, v, w take the 
values^, u x , v lf w 1 when x takes the value x lt we have 

y x -y = 4y> • • • a -y + 4y> 

and so for «, v, w. 

)\ — «i T" »! + U\ , 

y x -y = (u x -u) + (V x -V)+ (W 1 - W), 
or 

Ay — Au -j- Av + Aw. 
Ay _Au Av Aw 
Ax ~ Ax Ax Ax ' 

The student should observe the detail with which the difference- 
quotient is worked out here, as this detail will be omitted hereafter 
and he will be expected to supply it. 



44 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II. 

Since the limit of a sum is equal to the sum of the limits, we have 
for Ax( = )o, on passing to limits, 

By = Du + Dv + Dw, (I) 

or D(u -\- v ~{-w) = Du -f- Dv -\- Dw. 

Corollary. What has been proved for three functions here is 
equally true for any finite number of functions k, . . . u n , and it can 
be proved in the same way that 

£>2u r = %Du r ; 

hence the rule : 

The derivative of the algebraic sum of a finite number of differ- 
entiable functions is equal to the sum of their derivatives. 

In all cases in which we pass from an equation in difference- 
quotients to one in derivatives, the student is required to quote the 
corresponding theorem of limits, §15, which justifies the equality. 

EXAMPLES. 

1 . The derivative of any polynomial in x, 

a -f a x x + a 2 x* -{- . . . -f- a M x", 
is 

a x -\- 2a 2 x -j- 3a 3 Jf 2 -}-... -\-na n x n - 1 . 

This can be expressed in the following rule : 

Strike out every term independent of x, since its derivative is zero, and 
multiply each remaining term by the exponent of that term and diminish that 
exponent by I. 

2 . If y = 2x5 -j- log x b — 3 sin x, 
show that Dy = 5^ -f- 5/x — 3 cos x. 

3 . If /(*) =E CXl + b * + a , show that 

f\x) = c — ax-2. 

4. Make use of the identity 

sin (a -j- x) = sin a cos x -{- cos a sin x, 
to show that D sin (a -\- x) — cos (a -f- x). 

39. Derivative of a Product of Functions. 

Let y — uv. 

Then, with notation as in §40, we have 

Ay = (u -\- Aii){v -j- Av) — uv, 

= v Au -j- u Av -f- du • Av. 
Ay Au Av Au-Av ... 

Since, by hypothesis, 



Art. 40.] RULES FOR ELEMENTARY DIFFERENTIATION. 45 

are finite, the last term on the right of (i) has the limit o when 
Ax( = )o ; for it can be written either 

A i AV 

M — 



"© or (£') A, > 



and Au( = )o, z/z>( = )o, when zlv(=)o, the functions being con- 
tinuous. 

Therefore, in the limit, (i) becomes 

D(uv) = vDu + u Dv. (II) 

In particular, \iv is constant, v = a. then Da = o, and 

D(au) = aDu. 
Corollary. Show that 

D(uvw) = uv Dw -j- uw Dv -\- vw Du, 

and, in general, that the derivative of the product of a finite number of 
functions is equal to the sum of the derivative of each function multi- 
plied by the product of the others. 

EXAMPLES. 

1. Show that D{x n sin x) = nx n ~ l sin x -f- x n cos x. 

2. D{x* log x) = x*-* (log x a -f- 1). 

3. Show that £ {D log x sin x _ cos x i og x } — It 

x(=)o 

4. Show that D sin 2 x = sin 2x. 

2 

5. If y = (log x) 2 , show that Dy = log (-*")*. 

6. If /(*) EE log x 2 , then /'(*) = 2/r. 

7. Show that D sin 2x = 2 cos 2jt. 

8. Show that Z> cos 2jc = — 2 sin 2x. 

Use cos 2jt = (cos x -f- sin jc)(cos x — sin*). 

9. Show that D (log Jt:*) = log x -\- 1. 

40. Derivative of a Quotient. 

Let v = — . 

Thenj', «, z;, become y -f- z7>', « -j- z7w, z; -p- ^0, when x becomes 
x -\- Ax, and we have 

« -{- ^« u 

y ~ v -f~ Z/z> "" Z> ' 

z;z7# — #z7z> 



z>(z> -(- /fa) 



Au Av 

Ay _ Ax Ax 

Ax ~ v(v -f- Av) 



46 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ctt. II. 

Since Av( = )o when Ax( = )o, we have, provided v j£ o, on 
l)assing to limits 



In particular, if u = a, any constant, then Du = o, and 

Dv 

(IV) 



*o 



^ 



EXAMPLES. 

1. Show that D tan .r = sec 2 x. 

___ , _ _ sin x 

We have i? tan x = D , 

cos JC 

cos 2 jr -)- sin 2 •* 

cos 2 x 

= sec 2 x. 

2. Show that Z> cot x = — esc 2 x, using both 

cot .r = cos x/sin x and cot x = i/tan x. 

3. Show that D sec jt = sec x tan .r, 

using both sec x = i/cos x and sec x = tan x/sin x. 

4. Show that D esc jc =r — esc x cot x. 

5. Show that Z> vers x = sin x. 

6. Show that D , ~ = -. 

* + * (* + *)* 



'% 



8. Show that 



i 
log x log a x 



log ax " (log axf 

41. Derivative of the Inverse Function. — If y is a continuous 
function of x, we must have Ay( = )o when J.v( = )o, by the defini- 
tion of continuity. Therefore for any particular value of x at which 
y is a continuous function of x we can always make Ay converge to 
o continuously in any manner we choose, such that simultaneously 
we have Ax — o. Also, for corresponding differences Ay and Ax, 
we have 

Ay Ax _ 

Ax Ay ~ 

If we represent the derivative of v with respect to x, by D x y, and 
the derivative of x with respect to y, by D y x, then whenever y is a 
differentiate function of x and D x y ^ o, we shall have x a differ- 
entiate function of ;', and the relation 

D. x y.D y x = 1 
always exists. 

Therefore, if^ and x are functions of each other and the deriva- 



Art. 41.] RULES FOR ELEMENTARY DIFFERENTIATION. 



47 



rive of the first with respect to the second can be found, then the 
derivative of the second with respect to the first is the reciprocal or 
inverse of the first derivative. 

If v = f(x), then x = 0(.r), obtained by solving y = ftx) for x, 
is the inverse function of/"(~v). 



Geometrical Illustr 



vtiox. 



If the curve AB represents the function y = f(x), and we con« 
sider x as the function and y as the variable, we have x = (f)(y) 




Fig. 7. 

represented by the same curve, except that now Oy is the axis of 
the variable and Ox the axis of the function. For a particular x, 
the point X represents f\x) and <ft(y), and we have 

xX=/(x); yX=<f>{y). 

Again, if 6, are the angles made by the tangent to AB at X, 
with Ox, Oy respectively, measured according to the conventions of 
Cartesian Geometry, we have 

D x y =/ / (x) = tan 0, 
D r x = cf)'(y) = tan 0. 

But, since we always have tan 6 tan = 1, 















'• D x \ 


''DyX = I. 














EXAMPLES. 


1. 


If 


y = 


** + 


2ax 


+ A 


then 

DyX = 


2(x + a). 

1 




2(x + a) 


It 


we 


solve 


for x 


we 


get the 

x = 


inverse 
— a ± 


function 




Va? _l_ y _ b 



a function which we do not yet know how to differentiate, but we know its deriva- 
tive must be the value D y x obtained above. 

2. \iy = -T5, find D x y. D y x, and verify DyDx = 1. 



48 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II- 



3. Differentiate sin - l x. 

My = sin-'jr, then x — sin y 



.-. DyX = cosy — vi _ x 2 . 

Hence D x y — — . 

Vl - x* 

We know from Trigonometry that the angle whose sine is x, sin - *x, is 
multiple-valued and that 

sin [nit -f- (~i)"6] = sin 9, 

where n is any integer. In the derivative of sin -1 .* above, the radical shows its 
value is ambiguous as to sign. But if we agree to take sin— l x to mean that angle 
between — \n and -f \it whose sine is x, there is no ambiguity, since then cos y is 
positive. 

Then we have 

D sin— l x = . 



-f Vl — x* 
4. Show in like manner that 



D cos— x x = 



— Vl — x 1 
where o < cos— 1 x < n. 

This can also be shown immediately by differentiating the identity 
sin— l x -f- cos -1 .*- = \it. 

5. Show that D tan- 1 .* = \ * „ . 

I -\- x 2 

Put y = tan— 1 x, then x = tan y, and 

DyX = sec 2 y = i -f x 2 . Ex. I, § 4] 

.-. Z>ta.n-*x±=. +* 

where we take tan— l x to be that number such that 

— \it < tan- 1 * < -f \it. 

6. Show in like manner that 

D cot-Jx = 



1 4- x 2 ' 

where o < cot -1 * < it. 

Also, by § 38, from 

tan- 1 * -f- cot- x .r = \it. 

+ 1 



7. Show that D sec -1 * = 



x y'x 2 — 1 
If y = sec _I Jr, then x = sec y, and 



D y x = secy tan/ = x yx 2 — 1. Ex. 3, §41. 



x y'x 2 - I 
8. Show, as in Ex. 7, that 

D esc -1 * = 



x \/x 2 - I 
Also, by § 38, using the identity 

csc-'-r -f- sec- 1 * = lit. 



Art. 42.] RULES FOR ELEMENTARY DIFFERENTIATION. 49 

9. Differentiate a x . 

Put v = a*, then x = log a- y. 
Therefore, by § 34, we have 

DyX = - log a e. 

. -. D x y — = a* log, a. 

loga e 
In particular, if a — e, then 

Da x = a* log a 
becomes 

De* = <?* 
or the function e x is not changed by differentiation. 

42. Differentiation of a Function of a Function. — We come 

now to consider one of the most powerful methods of differentiating 
certain classes of functions.* 

Let z be a function of the variable y, say z =f(y), and let y be a 
function of Jf, say y = cp{x). We require the derivative of z with 
respect to the variable x. 

If is a differentiable function of the variable y, and r is a dif- 
ferentiate function of the variable x, for corresponding values of z, 
y and x, then we shall have 

D x z = D y z.D x y, (VI) 

or f x (y)=/;(y).I) x y. 

For, w^e have 

zte _ Az Ay 

Ax = Ay Ax' 
and since by hypothesis D y z and D x y are determinate limits, D x z is a 
determinate limit equal to their product, and (VI) is true. 

Corollary. If u is a function of v, v a function of w, w a function 
o(z, z a. function of_>', and finally ;' a function of ^, then the difference- 
relation 

Au _ Au Av Aw Az Ay 
Ax ~~ Av Aw Az Ay Ax 
leads to the derivative 

D x u = D v u-D w v-D z w- D^-D x y 
whenever the derivatives on the right are determinate. Hence the 
following rule: The derivative of a function of a function, etc., is 
equal to the product of the derivatives of the functions, each derivative 
taken with respect to its particular variable. 

EXAMPLES. 

1. Differentiate jr*, when x is positive and a irrational. 

Put j = x*, then taking the logarithm or, as we shall say, " logarating,"f we 
have 

logj' - a log x. 

*For a geometrical picture of a function of a function, see Appendix, Note 2. 
■f-The term "taking the logarithm" is the meaning of an operation so frequently 
used that it seems to deserve a verb "to logarate." 



50 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II. 

Differentiate with respect to x. We have 

-Dy = -. 
y x 

, \ Dy = a — = ax*-i, 
the same formula as when a is rational. 

2. Differentiate (a -f /;x) a . 

Put _A .' t = .)' a , where ;' = <7 -f- far. 

.-. f x (y) = ay«->Dy, and Z>j/ = b. 
.-. D(a -f bx)* = ba{a -\- bx)*-i. 

3. To find D cos x from D sin x = cos x. 
We have cos x = sin (-J-7T — x). 

D cos x = D sin (,V7T — .v). 

= COS (^7T — X) D(±7t — X), 

= — sin x. 

4. Deduce in like manner D cot x, D esc x, given the derivatives of tan x 
and sec x. 

5. If j = cos— J x, then x = cos;'. 
Differentiate Loth sides with respect to x. 

i = — siny Dy. 
.-. D cos— l x, as before. 

6. Find in like manner D cot- 1 ^, D esc— l x, from D tan x, D sec x. 

7. If y — a x , then log;' = x log tf. 
Differentiating with respect to x, we have 

D y log j • D x y = log a, 

or - D x y — log a. 

.-. D x y = a-* log a, as before 



8. Differentiate \ a 1 — x 1 . Put u — a 1 — x 2 . 
... Z>*«* = D u ii\D x u, 

= l«-2(- 2*), 



Va 2 - x 2 ' 

9. As an example of the differentiation of a complicated function of functions. 
differentiate 

log sin e cos(a-£*) 3 t 
Let 



y — a — bx. 

z = {a - bxf = y\ 

u = cos (a — bxf = cos z, 



w = sin e cos («—*•*) = sin z/, 
Therefore the required derivative is the function 

— vV' sin z cos v, 
w 

which can be expressed as a function of x directly. 



D x v = - />. 

ZV« = 3 f 2 - 

D z ii — — sin z. 

D H V rrr *«. 

Z>-,7<7 = COS Z\ 



Art. 43-] RULES FOR ELEMENTARY DIFFERENTIATION. 51 

43, Examples of Logarithmic Differentiation. — The differentia- 
tion of products, quotients, and exponential functions are frequently 

simplified by taking the logarithm before differentiation. 



EXAMPLES. 

1. Show that 

D(uv^) _ Du Dv 

1lV±i U V ' 

the upper signs going together and lower signs going together. Put^' = uv± l , 
then taking the logarithm, 

log y = log u ± log v. 
Dy _ Du , Dv 
y ~ it v ' 

This expresses compactly the formulae for differentiating the product and the 
quotient of two functions. 

2. Show that if u x u 2 . . . un is the product of n functions of x, the derivative of 
the product is given by 

DltfUr _ V^Dllr 
II«U r /j Ur % 

I 

3. Differentiate u v , where u and v are functions of x. 

Put y — uv and take the logarithm. 

••• log }' = v log «• 
Differentiating, 

Dy n . , Du 

= Dv • log u 4- v . 

y u 



Du* = K» (log u Dv -J Du]. 



4. Differentiate log„ u. 

Put r = log„ u, then v y = u. Logarate this with respect to the base e, and 
we have 

y log v = log u. 

Differentiating with respect to x, 

y „ Du 
log v Dy 4- — Dv = . 



D log„ u 



(Du log 11 Dv 



\ u log v v J log v 

44. For general reference in differentiation a table or catechism of 
the standard rules and elementary derivatives is compiled and should 
be memorized. 

In this table u or v is any differentiate function of a variable with 
respect to which the differentiation is performed. 



52 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. IT. 



The Derivative Catechism. 



i. D(cu) = c Du. 

2. D(u + v) = Du + Dv. 

3. D(uv) — u Dv -j- z>Z>#. 
z> Du — u Dv 



<-) 



v 4 
/i \ Z>z> 

^t) =--*■ 

6. Z># a = tf« a ~ : Du. 

Du , 

7. Z> log.w = — log a *. 

Z>« 

8. DXogu — . 

u 

9. Da u = # M log tfZ>w. 

10. Zte M = e u Du. 

11. Z>^ = «" log uDv -{- vu v ~ l Du. 

12. Z> sin z/ = + cos uDu. 

13. D cos zz = — sin #Z>#. 

14. D tan # = + sec2 uDu. 

15. Z> cot u = — esc 2 z/ Z?#. 

16. Z> sec « = -J- sec z/ tan ?/ Z)z/. 

17. D esc z/ = — esc u cot //Z>«. 

^" 

18. D sin -I « — -f- 



19. D cos~ l u = — 

20. D tan _I « = -f~ 

21. Z> cot _I w = — 

22. D sqc~ 1 u = + 

23. Z> csc _r z/ = — 

24. Z> vers _t z* 



Vi — z/ 
Du 

Vi — « 5 
Du 

+ 2" 

Du 

1 -f- «' r 
Z>/< 



z* 4/zz 2 - 
Z>/z 



7/ V/r - 1 
Du 

V211 — tr 



ART. 44-] EXERCISES. 53 



EXERCISES. 

1. Differentiate by the ab initio process, and check by the catechism, the follow- 
ing functions : 

(1), x. {2), ex. (3), 2*\ U),cx*. (5),*-'. (6),*r-4. (7), x* - 2X. (8), 
s.r 3 - 4_r -f 7. (9), l/(ax+ b). (10), x* - $x - 2x~*. (il), (x - l)( 3 x + 2). 
(12). (x - 3)/(.r + 5>. (13), *i (14), j'. (15), x~i. 

The solution in each case depends on the fact that a n — b n is divisible by a — b 

when n is an integer. 

x 
(16), cos-. (17), sin ax. (18), tan ax. (19), esc ax. 

2. Draw the curve y = 3X 2 and find the slope of the tangent where x = 2. 

3. Draw the curve;' — x 2 -f 2x — 3, and find the angle at which it crosses 
the Ox axis. 

4. Use the relation of the derivatives of inverse functions to find the derivatives 

of xK -ri x~K x~n , and check the results by the rule for differentiating a function 
of a function. 

5. Show that the equation to the tangent to any curve y = f(x) is 

y=f { a) + (x-a)f'(a), 
the point representing/^) being the point of contact. 



6. Differentiate Va* — x 2 , Vx 2 — a 2 , Va 2 -\- 2bx. 

Ans. - x(a 2 - x 2 )~i, x(x 2 - a 2 )~l , b(a 2 -+- 2bx)~i. 

(1) D.a -f xy = c{a -f xy-i. 

(2) D(a + x 2 ) 3 = 6x(a -f- a: 2 ) 2 . 

(3) D{c -\- bx 3 )* ■= \2bx\c 4. ^x 3 ) 3 . 

(4) Z>(ax 2 4- £x 4- cf — $(ax 2 4- bx + ^(2ax 4- 3). 

(5) Z>(a» - x 2 ) 5 = - \ox{d 2 — x 2 f. 

(6) D(a 2 x 4- &* 2 ) 7 = 7(« 2 x 4- bx 2 )\a 2 4- 2^). 

(7) D(b 4- or"*)* = mncx™-\b -f ftp«)»-i. 

(8) Z>(i -f ax 2 )' 1 = - ax(i 4- ax 2 )~K 

(g) D{a 2 - x*y = - \x\a 2 - x*)~K 

(10) D sin* ax = — D cos 2 ax = a sin 2ax. 

(11) Z> sin M ax — na sin^-^ax cos ax. 

(12) Z? sin (sin x) = cos x cos (sin x). 

7. Show that the equation to the tangent at x = a, y = ft, for the curve 

(1) x 2 4- y* = a 2 is xa 4"//? = fl2 - 

. . x 2 y 2 . xa yft 

(2) - ± ^ = 1 is — ± <-' - = 1. 

v ' a 2 b 2 a 1 b l 

(3) f 2 - \P X is J/ 3 = 2 /(* + a). 

8. Given sin 3X = 3 sin x — 4 sin 3 x, find cos 3*. 

9. Given cos 5X = 16 cos 3 x — 20 cos 3 x-f* 5 cos •*"> find sm S x - 

10. Verify cos x = 1 — 2 sin 2 |x, by differentiating. 

11. Obtain new identities by differentiating 

sin 3a 4" s i n 2a — sin a = 4 sin a cos |a cos fa, 
sin <5 sin (£# — b) sin (J7T 4~ ^) = i sin 3^> 
a and ^ being variables. 









54 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II. 

12. Differentiate the identity- 
cos 3 2x — 3 cos ix — 4(cos 6 x — sin 6 x). 



13. Differentiate x 3 sin x, x' 2 \/a -)- bx, (ax -\- by. 

14. D[{x + l) 5 (2* - i) :! ] = (\6x + i)(x -f l) 4 (2* - I) 2 . 

15. /ft* + !)(*■ - *)*] - 7 



2 j/x 3 — Jf 

16. Z?j(i - 2j-f 3x 2 - 4x 3 )(i + xfl = - 20jt- 3 (i 4- x). 

17. Z>{(i - 3JC 2 + 6**)(i + -r 2 ) 3 } = 6o^(i + a-)-'. 

18. Show that 



i + x _ i _ D x -f- 3 3 - 6x - jc- 2 



\ i - x (l _ x) ^ _ ^ x 2 + 3 (x 2 4- 3)-' 

x n nx n - 1 n x __3 



V 1 + x2 (i -f x 2 ) ? ' 3 + x* (3 + X 2 f ^ x 

\/a -\- x + \/a — x __ a 2 -\- a \/ a 2 — x 2 

\fa -\- x — 4/ a — x x l \/a 2 — x 2 
19. Show that 

x I _ . 3 + 2jc i 

JD tan-i ■ = — ; Z>sin-i">— i 



4/1 — x 2 4/1 — * 2 4/13 |/l - 3* — x 2 

2?. Differentiate sin— l (x/a), tan— '(ax -|« ^)> cos -1 - — , sec — 1 (a/x), 

4/a 2 -f- ■* 2 
sec- J (jr + ex 2 ). 

21. Z> log sin x = cot x; D log cos x = ? 

22. Differentiate e 2x , e~ x , e nx , e sinx , Jogx. 

23. Differentiate a cx , a s ™ ax , a l0 &* a****. 

24. Differentiate log x^, log (a -)- x), log (ax 4- /;), x M ^, a*ex. 2*, 
e* log (x + a), log (.*• -f- e x ), <?*/l°g ■*> ^°§ (■*'*)> s * n (**) * g ■*» *" cos * leg (cos x), 
log a tan x, $ lo e x , $ sinax , \og x 2 (cos ax). 

25. Z> sin [cos (ax -f- b) n ] = — na(a -f &r)*-i sin (ax -f £)» cos [cos(ax + b)"] . 

26. If j = £(** - *-*), show that 

^ = log ( y + 4/i +/>), 

and that jC^j Z^j: = 1. 

27. In Ex. 1, § 41, differentiate xasa function of>' and check the result there 
given. 



CHAPTER III. 
ON THE DIFFERENTIAL OF A FUNCTION. 

45. Definition. — The differential of a function is denned to be 
the product of the derivative of the function and an arbitrary differ- 
ence of the variable. 

If f{x) represents any function of x, and x\ — x any difference 
of the variable, then 

(-v, - *y(*) 

is the differential otf(x) at x. 

The value of the differential at a particular value x depends on 
the value assigned to the arbitrary number x v 

We use, after Leibnitz, the characteristic letter d to represent the 
differential, and write d/[x) to represent the differential of the func- 
tion J\x) at x. Thus 

df(x) = (x 1 - x)f\x\ 
=f'(x)Ax. 

46. Theorem. — The differential of a function is equal to the 
product of the derivative of the function into the differential of the 
variable. 

For, \ttf(x) = x, theny^jr) = 1, and 
dx = Dx • Ax 
= Ax. 
Therefore we can write dx for Ax, and have 

#T») =/'(•*) dx - 

The differential of the variable is then any arbitrary difference or 
increment of the variable we choose to assign. In writing the 
differential of a variable we choose to assign to it always a finite 
number as its value. In fact we cannot assign to it any other value. 

47. The Differential-Quotient of a Function.* — Since the differ- 
ential of the variable is a finite number we can divide by it, and have 

^-m 

* By some writers the derivative f\x) is called the differential-coefficient of 
the function /(jc), because of its relation to the differentials in the equation 

dffx) =/'(x)dx. 

55 



56 



PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. III. 



or, the differential-quotient of a function, which is the quotient of the 
differential of the function by the differential of the variable, is equal 
to the derivative of the function. 

This furnishes another notation, due to Leibnitz, for the value of 
the derivative, and expresses that number as the quotient of two 
numbers. The advantages of this notation will appear continually 
in the sequel, in the symmetry of the equations, and in the analogy 
and relation of differentials to differences. 

We frequently abbreviate the differential-quotient into 

df f , dv 

/-=/', or — , 
dx dx 

where y =zf(x). Also, \xhenf(x) is a complicated function we fre- 
quently write 

df(x) d 

dx 



s z*» 



48. Geometrical Illustration. — We have seen, § 31, that if 
y =/(x) is represented by a curve PP X , then the derivative/" (x) or 







J 


/ 


y 






^ 




p> 










M 








s$\ 









a 


a 


h 



X 



Fig. 



Dy is represented by tan 0, where 6 is the angle made by the tangent 
PT to the curve at P with the axis Ox. 

Assign any arbitrary number x xt and let P x represent f(x x ), and 
T the corresponding point on the tangent to the curve at P. Then 
we have 

PM —x x — x — Ax — dx. 
df(x) = (x x - x)f[x) 9 
= PM tSLTi MP1\ 

= MT. 

MT therefore represents the differential of the function f(x) at x 
corresponding to x v While 

MP, =/(.v,) -/(.v) = Af(x). 

rf/"and Af are more nearly equal when Ax or dx is a small number. 



f(x) = tan 6 



£ 



Art. 49.] ON THE DIFFERENTIAL OF A FUNCTION 57 

Observe that for a particular x the differential-quotient 
d/{x) 
dx 
is constant for all values of x v 

49. Relation of Differentials to Differences. — Since the differ- 
ence-quotient has the derivative for its limit, we can put 

<g =/'<-> + ., 

where a( = )o, when Ax(=)o. Therefore 

AJ\x) —f\x)Ax + a Ax, 

— f\x) dx -\- a Ax. 

. 4M -, I a 

' ' d/(x) + f'( X y 

Hence, when _/*'(.*•) 7^ o, we have 

4A*) _ . 

This substantiates the remark made in § 48 that the difference 
and the differential of a function are more nearly equal the smaller 
we take dx. 

50. Differentiation with Differentials.— Observe that all the 
formulae in the derivative table, § 44, are immediately true in differ- 
entials when we change D into d. For we need only multiply such 
derivative equation through by dx in order to make it read differen- 
tials instead of derivatives. 

We have 

d/[u) — D x /(u) dx. 
For, by definition, 

df{u) = DJ(u) du, 

= D u /(u).D x udx, 
= D x /{u) dx, 
since D x f(u) = f (u) D x u, and du = D x udx. 
.-. D u f(«)du = JD x f(u)dx, 

or the first differential of a function is the same whatever be the 
variable. 

More generally, let u, v, and w be functions of x. Distinguish- 
ing differentials like derivatives by subscripts, we have 

dj\u) = D v /(u) dv = D u f(u) D v u dv, 
— D u f{u) du = dj\u). 
In like manner, d w f(u) = d u f{u). Therefore 



58 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [ Ch - iil 

or the differentia] of a function is independent of the variable 
employed. It is not necessary, therefore, to indicate the variable by 
subscripts or in any other way; in fact the variable need not be 
specified. It is due to this that frequently the use of differentials 
has marked advantages over that of derivatives. 

51. We add a further list of exercises in differentiation, using in- 
differently the notations of derivatives, differential-quotients, and 
differentials in order to insure familiarity in their use. The sequel 
will show the advantage of each in its appropriate place, 



Art. 51.] EXERCISES. 59 



EXERCISES. 

1. If x, y, are the coordinates of a point on a curve, show that 

or ( V — y ) dx — {X — x) dy 

is the equation of the tangent at x, y, where X, Fare the current coordinates 
on the tangent. This equation can also be written 

Y - v X - x 



dy dx 

2. Show, with the above notation, that 

(F — y) dy -j- {X — x) dx = o 
is the equation of the normal at x, y. 

3. Show that d(x s logx) = x 2 (log x 3 -f- 1) dx. 

4. d(cos mx cos nx) =. — ;;/ cos nx sin mx dx — n cos mx sin nx dx. 

~ d . 

5. -7- sin M x = n sin"— 1 x cos x. 
dx 

6. d sin (1 -(- x 2 ) = 2 x cos (1 -\- x 2 ) dx. 

7. If y = sin'" x sin «, show that 

<#/ 

sin 2 x — = m sin w + J x sin(/;z 4- i)x. 
ox 

8. D{a sin 2 x -|- <$ cos 2 x) M = n{a — ^) sin 2x (a sin 2 x -f- ^ cos2 •*) M— *« 

9. d sin(sin x) = cos x cos(sin x) dx. 

10. f{x) = sin-i(jc w ) ( show/'(x) = «x m -i(i — x 2 *)-*. 

11. ^sin-i(i — x 2 )i = — (I —x*f^dx. 

d 3 4- a cos x ^a 2 " — b 2 

12. -r- COS- 



dx a -\- b cos x ~~ ' a -f- ^ cos x 

13. d sec M x = n sec M x tan x dx. 

idx 

14. a sec - J (x 2 ) = ■■ ■ . 

x Vx* — 1 

15. d(a 2 -f x 2 )i = x(a 2 + x 2 )-*</x. 

16. 4« 2 -^ 2 )-*z= x(a 2 - x 2 )~%dx. 

17. -^xrx 2 -f* 2 r* = 

18. D x (2ax - x 2 )* = (a — x)(2<zx — x 2 )-*. 

19. If /(x) = \x — \ sin 2x, then /'(-**) = sin 2 x. 

20. Show that </(-|x -f- ^ sin 2x) = cos 2 x dx. 

d /cos 3 x \ . 

21. -7- — cos x = sin 3 x. 

dx\ 3 y 

22. If j = sin x — l sin 5 x, then — = cos 3 x. 

23. d log cos x = — tan x dx. 

24. 7? log sin x = cot x. 

25. y = tan x — x. dy = tan 2 x dx. 



6o PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. III. 

26. 0(0 = cot t -f t, then (p\t) = — cot 2 /. 

27. If 2 = log tan \y, show that 



dz 

— — esc y = D log 

28. D m log tan ($jr + \m) = sec w. 



I — cos^ 

-y|i + cos^ 



«« ^ , /I + snl a 

29. ~j- log . — ■ — : = sec a. 

da " \| i — sin a 

30. r/sin-( 3 - 4 3 ) = 3(1 - 2 )~* ^0. 
d _, fl + * _ ^ gc ~ ^ 2 

<70 an ' j/ af _ & ~ CKpl + 2^0 4- C 

32. -j- log — - =-2 . 

du & \w -j- i/ « 2 — I 

„ d I i . (j + i) 3 . i , 2J — i 

33. — < - log ^-v- 1 H tan- 1 ^r— , 

ds I 6 g s* + i T ^T t / 3 J i + j 

34. tf log = = = — 

. C C |/ fl 2 -f C 2 

35. -^ | r/ V^^f 2 + « 2 sin- £ | = a |/F 

36. -r- sin 



\ « - /? 



^/ tf COS + ^ |/tf' J — <^ 2 

37. -7- cos • 



dz a -\- b cos 2 rt -|- b cos z' 

38. de*(i - x*) = e*(i - 3* 2 - X s ) dx. 

d (sin mv) n _ m« (sin mv) n —' 1 cos (w — ;//z/) 
~dv (cos nv) fn ~ (cos «7>/" r ' 

__ rf sin"' sin'"- 1 

40. — 7- « = 7~( m cos 6 + « sin 0)- 

rt'O cos" cos"+i v T ' 

41. <ftr* = x*(i + log x)dx. 

42. ^«* = ^x*(i -f log x). 

44 



/.r <?-* 4- *— * (,?* 4- e-*f 

45. </ log (** 4- e-*) - e J x ~ ^ - dx. 

46. De^+x) 2 sin a: = <?(a-f-*)' 2 [2(<z 4- x) sin x 4- cos x]. 
47 £ -JL- = "(i-*)-y 

<& ^ — I (** — I ) 2 

' dt \ Y + 4/T+lt*) ~ t j/F+ / 2 \i 4- t^ + 'V 



Art. 51.] EXERCISES. 

49. If xp{t) = <7 ( < l, - /S > , show that 

^(0 = a^-^~ k log a. 

(a» - r-f 

50. d tan a u = — a u u~ 2 sec 2 a * log a dtt. 

51. </ [9 + log cos (^tt - 8)] — 2(1 -f tan 0)- 1 </G. 

52. /?(# sin— *qfl) = sin— 1 ^ -f- ip(i — if> 2 )~*. 

53. Z>(tan tan-' 6) = sec 2 tan-* + (1 + 2 )-i tan 9. 

54. De~ a * xi cos £jc = - ^ 9jrS (2a 2 ^ cos &r 4- * sin for). 

1 1 

55. </*■* = x x (1 — logx) </jr. 

56. de e% = e* e x dx. 

57. Dx** = x** x*\x~* -f log x + (log xf\ 

58. <**** = xf X e* x-i (I + x log jt) </jt. 

59. Z>(i — tan jr) cos x = — cos jt — sin x. 

60. Z> log (log t) = 1 /log ft. 

61. If 0(0 = e 3t sin £/, show that 

0'(O -~ ** (« 2 + **)* sin (« 4- 6), 
where tan = bja. 

62. If sin^ = x sin (a -f _y), prove that 

dy _ sin 2 (# -\- y) 

dx ~ sin a 

63. If jt(i + yf + j(i + *)* = o, show that 

D x y = — (1 + x)-2 or lm 

64. If ^ =/(0 and x = ^(O* shovv that 

D x y = ^-. 

65. If xy = e*—y, show that 

dy log ar 



dx (1-flogjr) 2 

66. d (sin x) x = (sin jr)* (log sin x -\- x cot jr) (/jr. 

d „ log tan / 

67. ^ (log tan 0'=4-|^-. 



CHAPTER IV. 
ON SUCCESSIVE DIFFERENTIATION 

52. The Second Derivative. — The derivative/"^) of a function 
f{x) is itself a function of x, which is, in general, also differentiate. 

The derivative of the derivative f'(x) of a function f(x) we call 
the second derivative of/(;i-), and write \t/"(x). 

Thus 

xi{=)x 

For example, \lf(x) = x n , the first derivative /"(x) is nx n ~ l , and 
in the same way we find the second derivative 
f"(x) = 11(11 — i)x n ~ 2 . 
Again, \i/(x) = sin x, then 

f'(x\ = cos x and f'\x) = — sin x. 

If we use the symbol Df(x) to represent the operation of differen- 
tiation performed on f(x) } then two successive differentiations of 
f(x), which result in the second derivative, are represented by D'Y(x). 
... £[Z)f(x)] = Dy(x)=f"(x). 

EXAMPLES. 

1. D{a -f bx -f ex" 1 ) — b -f- 2<\r, 
Z> 2 (a + /;x -f fjr 2 ) = B{b + 2rx), 

= 2C. 

2. Z> cos <7jr = — a sin a.r, 

D 2 cos fl-Jtr = — aD sin rt.r = — a 2 cos tf.r. 

3. Z? log ax = <?/-r; Z>- log ax = — fl/jf 2 . 



4. D \fa 2 - x 2 = — x(a 2 - x 2 )~*, 

£T- tfa 2 — x* = — (a 2 - x 2 )-> -f x 2 :a 2 — x 2 )~%. 

53. Successive Differentiation. — The second derivative like the 
first is, in general, a differentiate function. Its derivative is called 
the third derivative of the function, and written 



/»,.,, ££& 



/"(*) 



Xl( = )X 



62 



Art. 54-] ON SUCCESSIVE DIFFERENTIATION. 63 

In general, if the operation of differentiation be repeated n times 
on a function /[x), we call the result the «th derivative of the func- 
tion. We write the »th derivative in either of the equivalent symbols 

D"f(x) =/»(*). 

It is customary to omit the parenthesis in f [u) {x), including the 
index of the order of the derivative attached to the functional symbol 
/"when there is no danger of mistaking it for a power, and write 

D«/{x) =/■(*). 

The index of either D or f in D" , f n denotes merely the order of 
the derivative and number of times the operation is performed. 

54. Successive Differentials. — In defining the first differential 
of a function, the differential of the independent variable was taken 
to be an arbitrary number. In repeating this operation it is con- 
venient to take the same value of the differential of the independent 
variable in the second operation as that in the first. In other words, 
we make the differential of the independent variable constant during 
the successive differentiations. 

Thus the second differential oif(x) is 

= d[/\x) dx], 

= d[f'(x)].dx, (i) 

since dx is constant. But, by the definition of the differential, 

«*[/»] = J3[/\x)l dx, 

=f"(x) dx. (ii) 

Substituting in (i), we have for the second differential 
d*f(x) =f"(x)(dx)\ 

or the second differential of a function is equal to the product 
of the second derivative into the square of the differential of the 
variable. 

It is customary to write the square of the differential of the 
variable in the conventional form dx 2 instead of [dx) 2 , whenever there 
is no danger of confounding 

dx 2 = (dx)* 
with d(x) 2 , the differential of the square of x. We shall write then 
dV(x)=f\x)dx\ 
In like manner for the third differential oi/[x) 
d\dy(x)-]=d[f'\x)dx*l ' 
= d[f"(x)].dx*, 
since dx is constant; and since by definition 

4T(*)] = D[/"(*V\ **, 

=/'"(x) dx, 



64 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV. 

we have for the third differential 

<^/(-v) =/'"(•*) JA 

and so on. 

In general, the nth differential of a function is equal to the 
product of the nth derivative of the function into the nth power of 
the differential of the independent variable. In symbols 

d n /{x) — f M (x)dx n f 

where it is always to be remembered that dx n means (dx) n , and d n , 
f n indicate the number of operations and order of the derivative 
respectively. 

EXAMPLES. 

1. We have d sin x = cos x dx. and 

d l sin x = fl\cos x dx) — d(cos x)-dx =. — sin x dx*. 

2. d*{a + bx 2 ) = d[2bx).dx — 2b dx 2 . 



3.d 2 \o g x=d^.dx = -^. 



55. The Differential-Quotients. — The nth differential-quotient 
of a function is the quotient of the nth differential of the function by 
the nth power of the differential of the independent variable. 

In symbols we have, from § 54, 

This symbol is also written, for convenience, in the forms 

ill of which notations are equivalent to either of 

/3"/(.v)=/»(.v), 
and are used indifferently according to convenience. 

56. Observations on Successive Differentiation. — In practice 
or in the applications of the Calculus we require, in general, only 
the first few derivatives of a function for solving the ordinary 
problems that are proposed. But, in the theory of the subject, i.e., 
the theory of functions, we are required to deal with the general or 
nth derivative of a function in order to know all the properties of the 
function. 

The formation of the nth derivative of a given function presents 
no theoretical difficulty, but owino: to the fact that differentiation, 
in general, produces a function of more complicated form (owing to 
the introduction of more terms) than the primitive function from 
which it was derived, the successive derivatives soon become so 



Art. 56.] ON SUCCESSIVE DIFFERENTIATION. 65 

complicated that the practical limitations (of our ability to handle 
them) are soon reached. 

The Differential Calculus as an instrument for investigating func- 
tions finds its limitations fixed by the complexity of the general or 
nth derivative of the function whose properties we wish to investigate. 

There are a few functions whose nth derivatives can be obtained 
in simple form, as will be shown below. 

We are aided in forming the «th derivatives of functions by the 
following: 

(1). The nth derivative of the sum of a finite number of functions 
is equal to the sum of their «th derivatives. 

(2). The «th derivative of the product of a finite number of func- 
tions can be determined by a formula due to Leibnitz, which we shall 
deduce presently. 

(3). The nth derivative of the quotient of two functions can be 
expressed in the form of a determinant and in a recurrence formula, 
directly from Leibnitz's formula. This is done in the Appendix, 
Note 3. 

(4). The nth derivative of a function of a function can be 
expressed in terms of the successive derivatives of the functions 
involved. This is also given in the Appendix, Note 4. 

In the application of the Calculus to the solution of ordinary 
geometrical questions, we need the first, frequently the second, and 
but rarely the third derivative of a function. When the function is 
given explicitly in terms of the variable, these derivatives are found 
by the direct processes as heretofore applied. If the derivatives are to 
be found from an implicit relation, such as <fi{x, y) = o, we can of 
course solve iory, when possible, and differentiate as before. It is 
generally, however, better to differentiate <p(x, y) with respect to x 
and then solve for Dy. If we wish D 2 y, we can either differentiate 
Dy with respect to x, or differentiate (p(x, y) = o twice with respect 
to x and solve the equations for D 2 y. 

In illustration, 

2JC 3 — 3j/ 3 — axy = o. 
6x 2 — ay — [gy 2 -\- ax)Dy — o, 
I2jc - aDy — (iSy Dy -f a)Dy — (gy 2 -f ax)D 2 y — O. 
Therefore 

6x 2 — ay 



D 2 y = 



Dy 

gy' -f- ax' 

I2x(gy 2 + ax) 2 — 2a(6x 2 — ay) gy 2 -f ax) — iSy(6x 2 — ay) 2 



( 9 y; _|_ axf 

Again, we frequently require the derivatives D x y and D£y, when we have 
given the polar equation 0(p, 6) = o, where x = p cos 6, y — p sin Q. 
We have 

D x y = D 9 yD x S=^-, 

_ sin 9 Dqq -f- p cos 8 

~~ cos 6 D e p — p sin 0* ^ ' 



66 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV. 

Also, 

D%y = D x {D x y) = D 'ZT x y) • D X Q, 

- D * y ' D&X ~ De} ' ' D o X 

p> -f 2 {D e P f -pDlp 
~ (cos 9 D e p — p sin S) 3 ' 
In which D e p and D\p must be determined from the polar equation (p(p, 0) = o.* 

EXAMPLES. 
1. Thewth derivative of x a , a being constant, 
(i). Let a = m be a positive integer. Then 

Dx m — mx m ~ I , 

D 2 x m = m{m — i)x™- 2 , 



J)n x m — m ( m _ i) . . . ( w _ w _j_ i)x>«-*, 
for all values of n < m. If n = m, then 

D m x m = m(m — i)... 3.2.1= m\ 
This being a constant, all higher derivatives are o. 
.-. D m +/>x m = o 
for all positive integers p. 
Also, when x = o, 

D n x™ — o, n < m. 

(2). Let the constant a be not a positive integer. Then, as before, 
D n x a = a(a — 1) . . . {a — n -f- i)jr a - w . 

Whatever be the assigned constant a, we can continue the process until n > a, 
when the exponent of x will be negative and continue negative for all higher deriv- 
atives. 

Consequently, when x = o, 

D n x* = 0, n < a. 

D*x* =oo, n > a. 

* The differentiation of an implicit function (p(x, y) = o is, properly speaking, 
the differentiation of a function of two variables, and a simpler treatment will be 
given in Book II. 

It will be shown in Book II that the derivative of y with respect to x, when 
0(x, y) = o, is 

dep 

dy _ dx 
dx = ~ djp ' 
by 

where -^ means the derivative of <p{x, y) with respect to x, x being the only vari- 
able ; ~ means the derivative of (p with respect to y, y being the only variable. 
dy 
For example, if <p(x, y) = 2x* — 2>>' — ax y — °> 

90 , , dip 

then ~ = 6x 2 - ay ; — - = - o^ 2 - ax. 

Ox oy 

Therefore, as in the text, 

dy _ 6x 2 - ay 

dx ~ oy 2 -+- ax ' 



Art. 56.] ON SUCCESSIVE DIFFERENTIATION. 67 

2. Deduce the binomial formula for (1 -j- x ) u t when the exponent n is a posi- 
tive integer. 
We have 

(1 + x)(i -f x) = (I -f af = i + 2x + **, 
(1 + x)(i + *)« = (f + *)P = 1 + 3* + 3 *a 4- «•; 

By an easy induction we see that (1 -f .r) w must be a polynomial in x of degree 
n. It is our object to find the numerical coefficients of the various powers of x in 
this function. Let 

(I + x)» = a -f- a x x -f a^a + • • • + ««*«. 
Differentiating this r times with respect to x, we have 
»(«— I) . . . («— r-(-l)(i+jr)»- r =r! a r -f- . . . -f- n[n— 1) . . . (n—r-\-i)a n x H ~ r . 
This equation is true for all assigned values of x and r, and when x = o, 
«(» — 1) . . . (« — r -j- i\ 

/Z_ — - /. 



a number which it is customary to represent conventionally by either of the 
symbols 



P> 



This number is of frequent occurrence in analysis. In Algebra, when n is an 
integer, it represents the number of combinations of n things taken r at a time. 
Hence we have the binomial formula of Newton, 

(I +*)« = J C n xr. (I) 

r = o 

Corollary. If we wish the corresponding expression for (a -\- y) n , then 

{a+yy = a* (1 + ^V- 
Put j/a for x in (1), and multiply both sides by a n . 

... {a+y)" = 2 C M r a«-rxr. 

r = 

This can be written more symmetrically thus: 

(a -f j/)» _ * a«-r a:'' 
« ! — ^ (n — r , ! r f 

o V ' 

3. The wth derivative of log x. We have 

D log x — — = .r-i. 
Therefore, by Ex. 1, 

Z>» log jr = (— i) w -i(» — 1) ! — . 

4. The «th derivative of a*. We have 

Da x = a x log a. 
. •. D^a* -- a x (log a) n . 

In particular, Zte* = e x \ D n e x — ^*. This remarkable function is not changed 
by differentiation. 



68 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV. 

5. The nth derivative of sin x and cos x. 
We observe that 

D sin x = -j- cos x\ D cos x = — sin x; 

D 2 sin j: = — sin x\ D 1 cos .r = — cos x; 

D* sin x = — cos x; Z> 3 cos x = -f- sin x\ 

D k sin x = + sin ;r; Z> 1 cos jr = -f- cos jr. 

Thus four differentiations reproduce the original functions and therefore the higher 
derivatives repeat in the same order, so that 

D±n-\ sin x = (— i)*- 1 cos x; D 2H ~ l cos x = (— I) M sin jr; 

D 2n sin a: = (— l)« sin jc; Z> 2M cos # = (— ij" cos x. 

In virtue of the relations 

cos x = sin [lit -\- x), sin jt = — cos (\it -\- x), 

these formulae can be expressed in the compact forms 

D n sin x = sin [ # -| # 



Z>" cos x = cos f jc -| 
6. Given ^ +4 = I > find ^7,^7- 



n \ 



Differentiating with respect to x, 
Differentiating again, 



x y ay 

|_ <- jL — o. 

<z 2 ^ £ 2 ^ 



d^ b\dx) ^ b 2 dx 2 ~ ' 

* • tflr 2 ~~ ; J a 2_h ^^y f ' 

^* . o> V- x 

— — , since — = - , 

ay 6 dx a 1 y 

Differentiating again, we can find 

d' A y -$b*x 

dx 3 ~ a*y* 

7. If y* = 4<zx, show that 

dy za d-y 
dx~ y ' dx 1 ~~ 

8. If y' 1 — ixy = a 2 , show that 

dy y d-y a 1 

dx ~ y — x" 1 dx' 1 ~ (y — jc) 3 

9. From the relation jr 3 -j- j 3 — 2> ax y — °? 

dy _ x- — aV d 2 y 

dx ~ y 2 — ax ' dx* " ~ (j 2 — ax) 3 ' 

10. If sec x cos j = a, show that 

dy _ tan x dy _ tan 2 v — tan 
dx tan 7 ' dx 2 ~ tan 3 j 



w 

yf 






dy 


3 a 


2 x 


dx*~ 


(y- 


*f 


show that 






zcPxy 





Art. 57. J ON SUCCESSIVE DIFFERENTIATION. 69 

57. Leibnitz's Formula for the nth Derivative of the Product 
of Two Functions. — Let //, v be any two functions of .v. For sake 
of brevity, let us represent the successive derivatives of u and v by 
these letters with indices, thus : 



u , u , «,..., w 

v', v" ', v'" 9 . . . , v" 



Then 



D{uv) = u'v -j- v'u, 
D 2 (uv) = u"v -f- u'v' -f- u'v' -f- v"u, 
— 2i"v -J- 2«V -f- 7/z>". 

In like manner, differentiating again this sum of products, we find 
on simplification 

B 3 (uv) — u'"v -j- 3«'V -f 3«V + «z>'". 

Observing, when we use indices to indicate the derivatives, the 
symbols LPu, f°(x), v°, mean that no differentiation has been 
performed and the function itself is unchanged, 

. • . JDPu = u° = u, and f°(x) =/(x). 

In the above successive derivatives of uv we observe that the 
indices representing differentiation follow the law of the powers of 
u -f- v when expanded by the binomial formula, and the numerical 
coefficients are the same as those in the corresponding formula of that 
expansion 

In order to find if this law is generally true, let us assume it true 
for the nth derivative and then differentiate again to see if it be true, 
in consequence of that assumption, for n -j- 1. 

Assume that (see Ex. 2, § 56) 

n 

D n (uv) = 2C Mtr u n ~ r v% 

r = o 

= u n v +C HtT u n ~ l v'+ ...+C Mtr u n ~ r v r + . . . + uv n . 
Differentiating this, we have 
&*+ 1 (uv)=u n+I v+C„ tJ «V+...-fC M)r u n - r+l v r +...-j-u / v" 

_j_ u n v , +...-\-C njr _ 1 u n - r+l v r -\-...+nu'v n +uv n+ \ 

= t^^v^C n+ltl u-v'-\-...-\-C n+l<f .u^-^+...+uv^\ 

r = 

in virtue of the relation* C n<r -\- C„ ir _j = C n+lr . 

Therefore, when the law is true for any integer n, it is also true 
for n -j- 1. But, being true for n = 2, 3, it is true for any assigned 
integer whatever. 

* n\ _j n\ _ n\ / 1_ I \ _ (n + i)\ 

r\{n—r)\ ' (r— i)!(»-r-f-i)! ~~ (r-i)\{n-r)\ \r ~*~ n-r+i) ~ r\{n+i—r)\ 



70 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV. 

We can express the nth. derivative of the product iw symbolically, 
thus: 

D n (uv) = (u + v) n , 
in which (u -f- v)" is to be expanded by the binomial formula, and 
the powers of // and v in the expansion are taken to indicate the orders 
of differentiation of these functions. Remembering that when the 
index is a power we have u° — i, but when it means differentiation, 
u° = u. 

EXAMPLES. 

1. To differentiate the product of a linear function by any function f{x). 
Let u =z {ax + b)f K x). 

Then D(ax -f b) — a, D\ax + b) = o, 

.-. D n u — {ax + b)f n (x) -f- nafn—^x). 

2. In like manner show that the nth derivative of the product of a quadratic 
function of x, say y, by any other function /, is 

yfn _|_ ny'fn-i _|_ £»(« — l)/ '/"-*. 

3. Show , if (p(x) and ip{x) are differentiable functions of x, 

Dn[0ix)ib(x)-\ _ * <p»-r(x) lf>r( X ) 
n\ ^ (n - r)\ r\ ' 

58. Function of a Function. — A formula for the «th derivative 
of a function of a function will be deduced in the supplementary 
notes.* However, the simple case of a function of a linear function 
of the independent variable is so useful and of such frequent occur- 
rence that we give it here. 

Let u = ax-\-b, and f{u) be any differentiable function of u. Then 

D x f{u)=f u { U )D x u, 

= dfju). 
Dl/(u) = aD x [f u {u)] i 

= af' u \u) Du, 



and generally 



1. Show that 



2. D"e ax = a n e ax 

3. Show that 



I) x /(u) = *"/». 

EXAMPLES. 

D n sin ax = a M sin (ax -\ 7t\ 

£>" cos ax = a n co. c [ax -I 7t\. 

\ 2 / 

D n (-±-\ - W! 

"" \x - a) ~ (x - «)»+i 



Appendix, Note 4. 



_, 



Art. 58. 1 EXERCISES. 71 



EXERCISES. 

Show that 



\x) x*+i 

\x») K ' x n + r 

3. DA = r\ — - . 

\C - x) \C - X)r+i 

4. D« log (1+*)= (- i)-« |* ~^ . 

5. Z> 4 (x 3 log jr) = 6.X-1. 

6. /^(jc 4 -(- <z sin 2jc) = 32a cos ix. 

7. Z£(j«*) = x D r u 4- r D r ~^n, where u is any function of x. 

8. ££(<* — •*•)« — {a — x) Dm — r Dr—iu. 

9. /^(x*- log x) = - 4 ! x-2. 

10. Z> M (* log *) = (- i) n (n — 2)! jr-*+i. 

11. £> 2 .r* - ^(1 -f log Jf) 2 + #»-i. 

12. Z> 3 log (sin .r) =: 2 cos x esc 3 .r. 

13. Z^JcMog.r?) = 2±.r-i. 

14. Z?*a« = a* (log a c ) M . 

./ ! { ac -\- b ac — b 

15 



X* - C* K ' 2C \ {X — C)n+i 



(X + ,)«+ 

Observe that by the method of partial fractions we can write 
ax -J- b I $ ac -\- b ac 



(x — c)(x -\- c) 2c I x — c x -\- c 

ax -\- b _ ^ 1 l ap + £ _ ^_+_^\ _ . 

(jf - /)(* - q) p -q\x—p x - q) 



17. Make use of the method of partial fractions, to find the «th derivative of 

ax i 4. bx + c _ I t af+pb-\-c aq 2 + bq -\- c \ 
(x - p){x - q) = / - ? ^ x - p x - q J + d 

_ _, , I d\*2x 2 — 4_r— 6 , «! 

18. Show that — ] — ^ = - 6 — - . 

\y.r J x 2 — $x -f- 6 (2 — x) M + I 

19. If j = fl(l + x 2 )-i, show that 



(1 + x 2 )y( n ) 4- 2nxy( H -V 4- «(» — i)_y(*- 2 ) = 

20. If f(x) = a cos (log x) 4- b sin (log x), show that 

x 2 f"{x) + xf\x)+f(x) = o. 

21. Show in 20 that the following equation is true : 

x 2 /«+2 4- x(2n 4- i)/«+i 4- (« 2 4- i)/ n = O. 

22. If ;/=*<* sin -1 * show that 

(1 — jc 2 )</ 2 _y — x dy dx = a' l y dx 2 , 
thence find, as in 21, an equation in>"+ 2 , y n+1 , y n - 



72 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cn. IV. 

23. If y = (x + IV 2 - i)"\ show that 

(x 2 — i)d 2 y + x</j' ^ — m 2 y dx 2 = o. 

24. If ^ = sin(sin x). show that 

</ 2 ^ -f- tan x dy dx -j- y cos'jt dx 2 = o. 

25. If y = A cos «x -f- .5 sin «.r, then 

D l y -\- ft-y = o. 

n 

26. Z?**** cos ^jt = (a 2 -f- b 2 ) 2 e ax cos (dx -f- »0), where tan = J/a. Differ- 
entiate once, twice, and observe that the law follows directly by induction. 

27. D H tz.n-*x-* = (— i)«(» — i)! sin* (tan— i#— ») sin »(tan-**r-s). 

Put j = tan— i jt~ i . Then x = cot/, and 

Z^ = — (I -f- x 2 )- 1 = — sin 2 j. 
D x y = — D x sin 2 y ='— D y sin 2 y Dy = sin 2 y sin 2y 
The rest follows by an easy induction. 

28. If x = (p(t) and y = ^(/), then/ is a function of x 
Required D x y. D x y> 



We have 


dy — ip'(t)dt, dx = <pt[t)dt. 




dy #(/) 




dx 0\t) ' 


Also, 






d 2 y d Tp t ' d ibt dt 




dx 2 ~ dx (pi ~~ dt <pt ax 




il>t"(pt — il>t<t>t" dt _ 




(4>tf dx , 




d 2 y dx dy d 2 x 




~di 2 ~di ~ "dt ~dt 2 




m ' 


29. If x = 


: sin 3/, y = cos 3/, show that 




Dly = - y~3. 


30. Z>» tan- 


... , sin (n tan- 1 .*— 
- J x — (— 1 )«-x(» 1) ! v . ; 



<pt 



(I + **f« 

This follows immediately from Ex. 27, since 
tan-ix = \it — tan-ijc- 1. 

31. If v = tan-'jr, show that 

(1 -f x 2 )y(»+*) -f 2nxyi*) -f n(n - i)j/(*-i) - o. 

32. D»(a 2 -f x 2 )-i = (— i)«« ! a-«-2 s in*+i0 sin (» -f i)0, 
where tan = «/.*■. Hint. Use Ex. 30. and 

D tan-i(a/x) — — a(d 2 -f x*)~*. 

33. Z)"x(a 2 4- x 2 )-i = (_ i)«a-*-i» ! sin»-H0 cos (n + i)0 
where tan <p = a/x. Use Ex. 32 and Leibnitz's Formula. 

34 - "7- = 2(— i)» . 

\dx i+x V ; (1 + *)«+i 



Art. 5S.] EXERCISES. 73 

_ 2 V.T - 1 nt n 
35. /)V- *& = 



2.1- t .1 

36. If .)'-'( 1 + -*" 2 ) = (I — x -f x 2 )-, then 

<*!? ' _ I + 3* + -*" 2 

ak 2 ~ (i_}_ ^2)i 

37. If j = sin (w sin— ! x), prove 

("«') 2" -*£■+-?»'=■* 

38. If y = sin-'.r, deduce 

(I _ x i )y " _ X y' - o, 

and (1 - x*) — 2. t 2n + i)x- } - n 2 -^ = o, 

by applying Leibnitz's Formula to the above. The deduction of such differential 
equations is of fundamental importance for the expansion of functions in series. 

39. Show that 

Apply Leibnitz's Formula to the product/^) •(« — #)— 1. 

40. Show that 

where, in the differentiation indicated by — - , x is constant and y the variable. 

ay 

The result follows at once when Leibnitz's Formula is applied to the product of the 

two functions_/(jr) — f(y) and (x — y)— I . 

This is one of the most important formulae in the Calculus. Observe that it is 

obtained by successive differentiation of the difference-quotient. 

41. Show that the derivative of the right member of the equation in Ex. 40, 
with respect to y (x being considered constant during the operation), is 

_(^L / „ +I( ,, 

Hint. Differentiating each product in the sum, we find that the terms all can- 
cel out except the last. 



CHAPTER V. 
ON THE THEOREM OF MEAN VALUE. 

59. Increasing and Decreasing Functions. 

Definition. — A function f(x) is said to be an increasing func- 
tion when it increases as its variable increases. A function is said to 
be a decreasing function when it decreases as its variable increases. 

In symbols, f(pc) is an increasing function at x = a when 

f(x) -/(a) (1) 

changes from negative to positive (less to greater) as x increases 
through the neighborhood, (a — e, a -\- e), of a. In like manner 
f(x) is a decreasing function at a when the difference (1) changes 
from positive to negative (greater to less) as x increases through the 
neighborhood of a. 

60. Theorem. — A function f(x) is an increasing or decreasing 
function at a according as its derivative f'(a) is positive or negative 
respectively. 

Proof: If f(x) is an increasing function at a, the difference- 
quotient 

A*) -A«) 

x — a 

is always positive for x in the neighborhood of a, consequently its 
\\m\X.f'{a) cannot be negative. If /'(a) is a positive number, then 
for all values of x in the neighborhood of a the difference-quotient 
must be in the neighborhood of its limit f'(a), and therefore posi- 
tive. The function is therefore increasing at a. 

In like manner, \i/(x) is decreasing at a, the difference-quotient 
is negative for all values of x in the neighborhood of a and therefore 
its limit cannot be positive. Hence, if /"'(#) is a negative number, 
the difference-quotient must be negative for x in the neighborhood 
of a, and therefore _/*(.*) is decreasing at a. 

Geometrical Illustration. 

Let/ = f(.r) be represented by the curve A y 4 r The function is increasing at 
A x and decreasing at A r 



We have 
for 6 t is acute, while 



f'(a x ) = tan 9, = +, 

f'i'h) = ^n 2 = -, 



74 



Art. 6i.] 



ON THE THEOREM OF MEAN VALUE. 



75 



since 6 2 is obtuse. Remembering that, under the convention of Cartesian coordi- 
nates, the angle which a tangent to a curve makes with the .r-axis is the angle 
between that part of the tangent above Ox and the positive direction of Ox. 



V 


E 


s-^2 




Y^ 


^\ 




,A 




V 





/ <*1 


a 2 



Fig. 9. 

61. Rolled Theorem. — If a function f(x) is one-valued and 
d Afferent i able in (a, /?), and we have f(ac) =f(fi), then there is a 
value B, of x in (a, fi) at which we have 

/'(£) = o, 

provided f"\x) is continuous in (a, /?). 

If fix) is constant in any subinterval of (a, /?), its derivative 
there is o and the theorem is proved. 

lif(x) is not constant in (a, /?), then at some value x' in (a, ft) 
we shall have J\x') ^/(a). If J\x') > /{a) = /(/?), the function 
must increase between a and x' and decrease between x' and /?, in 
order to pass from /"(a:) to the greater value/j^r'), and from/^') to 
the lesser value /[/3). Also, if /(•*') </(«) =/(/?), then the func- 
tion must decrease in (a, x') and increase in (x' , /?), for like 
reasons. In either case the derivative f\x^) at some point x x in 
(#, x') must have contrary sign, § 60, to the derivative f'(x 2 ) at 
some value x % in (x ; , /?). 

Since f\x^) and f'{x^ have opposite signs, and f'(x) is, by 
hypothesis, continuous in (pc A , x 2 ), then there is, § 23, I, a number 
B> in (x 1 , x 2 ), and therefore in (a, /?), at which we have 

/'{£) = o. 

In particular, if /{a) = o and/(/?) = o, then there is a number 
B, between a and fi at which 

f'(S) = o. 

Rolle's Theorem is usually enunciated : If a function vanishes for 
two values of the variable, its derivative vanishes for some value of 
the variable between the two. Or, the derivative has a root between 
each pair of roots of the function. 

The figure in § 60 illustrates the theorem. 

62. Particular Theorem of Mean Value. — If f(x) is a one- 
valued differentiate function having a continuous derivative in 
[a, /?), and if a and b are any two values of x in [a, /?), then 

fyl) -/(a) = (b- a)/'{S), 

where £ is some number in [a, b). 



7 6 



PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. V. 



The truth of this theorem follows immediately from Rolle's 
Theorem. 

Let k represent the difference-quotient 

A") 



k=m 



Then 



or 



b-a 



A*) -A«) = (* - «)*> 



f{b)-kb=f(a)-ka. (i) 

The function f{x) — kx is equal to the number on the left of 
(i) when x = b, and to the number on the right when x = a. 
Therefore, by Rolle's Theorem, having equal values when x = a and 
when x = b, its derivative must vanish for x = B, between a and b. 
Differentiating, f{x) — kx, k being independent of x, we have at £, 

/'(g) -k = o, 
which proves the theorem. 

Another way of establishing the result is to observe that the 
function 

(a - b)/(x) - (x - b)f(a) + (.v - a)/(i) 
vanishes when x = a, also when x = b. Therefore its derivative 
must vanish for some value of x, sav £,, between a and b. 
.-. (a - i)f'{£) -f(a) +f(i) = o. 



Geometrical Illustration. 


y 




? 




i 


M 




%^ 


B' 






A' 


X' 








~*^\ 













c 


i i 


V 


( 


) 



Fig. io. 

Each of these processes admits of geometrical illustration. 

(i). k is the trigonometrical tangent of the angle which the secant AB makes 
with Ox. Draw OA'B' parallel to AB. Then 

BB' = /(/;) - kb = A A' - /[a) - ka. 
XX' = f{x) — kx is equal to AA' when x = a, and to BB' when* == 6. The 
theorem asserts that there is a point E on the curve y ~ J\x) having abscissa % at 
which /'(£) = k, or the tangent at E is parallel to the chord AB. 
(2). The function 

(a - d)/(x) -(x- b)f{a) + (x - a) /(b) 
is nothing more than the determinant 

/(->-). x, I 
/(a), a, I 
/(/')• b, i 



Art. 63.] ON THE THEOREM OF MEAN VALUE. 77 

which is the well-known formula in Analytical Geometry for twice the area ol the 
triangle AXB, in terms of the coordinates of its corners. This vanishes when X 
coincides with A or B. It attains a maximum when the distance oJ X from the 

base AB is greatest, or when X is at E, where the tangent is parallel to the chord. 
This theorem amounts to nothing more than Rolle's Theorem when the axi - of 
coordinates are changed. 

63. Lemma. — Ex. 39, § 58, forms the basis of the most important 
theorem in the Differential Calculus, i.e., the Theorem of Mean Value 
for a function of one variable. On account of its usefulness, we inter- 
polate its solution here. 

The starting point of the Differential Calculus is the difference- 
quotient. On that is based the derivative of the function. We shall 
now use it in presenting the Theorem of Mean Value. 

Let/(x) be a one-valued successively differentiable function of x in 
a given interval (a, (3). Let x represent any arbitrary value of the 
variable, and y some particular value of the variable at which the 
derivatives of/~are known. 

(1). Consider the difference-quotient 

A*)-M 

x — y 

If we hold x constant while we differentiate this n times with 
respect to the variable y by Leibnitz's Formula, § 57, and then 
multiply both sides by 

(•* ->r +1 

n\ 

we shall obtain 

/(-v) -Ay) - (* -y)f\y) ----- ^^>W 

= {*-y) a+1 (*\" (A*)-Ay) \ 

For, we have 

*>;IA*) -Aril = -fh')> 

D;- r (x -y)- 1 = (n - r)\(x - y y-<"+», 

which values substituted in the form of Leibnitz's Formula in Ex. 3, 
§ 57, give the result. 

(2). On account of the importance of this formula we give 
another deduction which does not use Leibnitz's Formula directly. 

Let 

A*) -Ay) = Q 

x-y 

Then 

A*)-Ay)±(*-s)Q- 



78 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. V. 

To introduce the known derivatives at y, let x be constant and 
differentiate this last equation successively with respect to_>'. Thus 

-/X>') =(*-jOg_a (2) 

-f'\y) = (* ->')«' - *& (3) 

-/«0) = (* -j^)(2t M) - «^- 1} . (»+ 1) 

Multiply (2) by (a- -jy), (3) by i (* - j^ . . . , and (a + 1) 
by — (x — y) n , and add the n -f- 1 equations. There results 







/(-v) -/w - (*-j-)/*w-. • • - ^r-/"0) = ~^' <?*• (q) 

the same formula as in (1). 

64. The Theorem of Mean Value. Lagrange's Form. — The 

Theorem of Mean Value, which we now present, is the most impor- 
tant theorem in the Differential Calculus. The applications of the 
Differential Calculus depend on it as do also its generalizations. It 
is but a direct modification of the differential identity (q) established 
in § 63, and consists in the evaluation of the nth derivative, Qf\ of 
the difference-quotient Q in a different form. 

Consider the arbitrarily laid down function of z, 



(*-*)* *w.\ ( x -*y 



H 



)(«) 
y > 



m =/(*) -A») -(*-My (*)-...- i-j^/'W - '' n ,' Q 

in which, as in § 63, 

does not contain z and is constant with respect to z. 

Observe that this function F(z) is o when z = x, because the first 
two terms cancel and all the others vanish when z = x. Also, F(z) 
is o when z =y, by reason of the identity (g). 

Consequently, by Rolle's Theorem, § 61, the derivative F'(z) 
must be o for some value B, of z between x and y. Differentiating 
with respect to z, and observing that the terms on the right, after 
differentiation, cancel except the last two, we have 

*'(«) = - {jL ^r 1 f ,+ \*) + (» + ^r 1 ^- 

Hence, when z = £,, at which F'(£) = o, 
V * ~~ n + 1 * 



Art. 65.] ON THE THEOREM OF MEAN VALUE. 79 

Substituting this value in (q), we have Lagrange's form of the 
Theorem of Mean value,* 

/(•v)=A.)+(.v-t')/'0')+- • .+ (£ =r ! /"W + { -^^f M ( s )' 
Sr-^+V+V 7 " 1 ^- (L) 



=1 



65. Theorem of Mean Value. Cauchy's Form. — Cauchy has 
given another form to the evaluation of the difference 

r = o 

which for some purposes is more useful than that of Lagrange. Its 
deduction is somewhat simpler. 

Let x be constant and z a variable. Consider the function 

F(z) =/{z) + (x- z)f\z) + ... + (X ~ Z) "/ "(z). (i) 

By the Theorem of Mean Value, § 62, 

F(x) - F(a) = (x- a)F'(S), (ii) 

where B, is some number between x and a. 
When z = x, we have from (i) 

F(x) =/{x). 

When z = a, then from (i) 

F(a) =/(a) +(x- d)/\d) + ... + {X ~ a) " / "(")■ 

Differentiating (i), 



and 



F,'{*) = £-^/»H>>), 



F'(S) = ^—fil/^S). 



Substituting in (ii), we have Cauchy's form 

n 

/{x) = J^ (f-=^L r{a) +{x - a) { -^±y^(£). (C) 

r = o 

* In order that this result shall be true, it is necessary that the function f{x) 
and its first n -f- 1 derivatives shall be finite and determinate at x and at^', and 
also for all values of the variable between x and^. This important formula will 
be presented in another form in the Integral Calculus, Chapter XIX, § 152. 

For a proof of the Theorem : If a function becomes 00 at a given value of the 
variable, then all its derivatives are 00 there, and also the quotient of the deriva- 
tive by the function is 00 , see Appendix, Note 5. 



80 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. V. 

The numbers represented by B, in (C) and in (L) are not equal 
numbers. All we know about S, in either case is that it is some 
number between certain limits. 

66. Observations on the Theorem of Mean Value. — The formula 
(L) or (C) is a generalization of the theorem of mean value stated 
in § 62; that theorem corresponds to the particular value n = o. 

The Theorem of the Mean is the basis of the expansion of a 
function in positive integral powers of the variable. When this 
expansion in an infinite series is possible, it solves the problem : 
Given the value of a function and of its derivatives at any one par- 
ticular value of the variable, to compute the value of the function 
and of its derivatives at another given value of the variable. 

The Theorem of Mean Value is the basis of the application of the 
Differential Calculus to Geometry in the study of curves and of sur- 
faces, as will be amply illustrated in the sequel. 

It solves the problem : To find a polynomial in the variable which 
shall have the same value and the same first n derivatives at a given 
value of the variable as a given function. This polynomial, therefore, 
has the same properties as the given function at the given value of 
the variable, so far as those properties are dependent on the first n 
derivatives. This is a most important and valuable property of the 
formula, for it enables us to study a proposed function by aid of the 
polynomial, and we know more about the polynomial than about any 
other function. 

67. In Chapters I, . . . , IV, we may be said to have designed 
the tools of the Differential Calculus, for functions of one variable, 
in the derivatives on which the properties of functions depend. 

In the present chapter this design may be said to have culminated 
in the presentation of the Theorem of Mean Value. 

The subject has been developed continuously and harmoniouslv 
from the difference-quotient. The difference-quotient is the founda- 
tion-stone from which the derivatives have been evaluated, and by 
successive differentiation of the difference-quotient we have been led 
to the Theorem of Mean Value. 

It is not necessary to add here any exercises or examples of the 
application of the Theorem of the Mean, since it will be employed 
so frequently in what follows. We merely notice other forms under 
which the formula may be expressed. 

68. Forms of the Theorem of Mean Value. 

(1). It is customary to write R n as a symbol of the difference 
between the functions 



n 

f{x) and £<*^*T/'(a), 



Art. 68.] ON THE THEOREM OF MEAN VALUE. Si 

so that 

n 

o 

Or, more briefly, 

where S H represents the 2 function. 

(2). In particular, if a = o, and f(x) is differentiable, n -\- 1 
times at o and in (o, x), we have 

/(-*) =/(o) + .v/'(o) + . . . + J/»(o) + ie. , 

where, using Lagrange's form, 

R « = (£pi)/" +, <*>' * in (°> *>• 

or, using Cauchy's form, 

R = X ( X ~^' % /»+U£), £ in (o, x). 

(3). If we write the difference x — y — h, so that x =zy -\- h y 

/O + h) =/0) + a/'O) + • • • + J/*O0 + *«■ 

(4). Again, since h is arbitrary we can put h = dy. Then 
/ty + 60 =f(y) + d/(y) + ... + ^M +J fi>„, 



or 



„=*+%+._..+%+*. 



EXERCISES. 

1. If f(x) = o when .# = «j , . . . , x '— a n , where 
a x < a 2 < . . . < a H , 

and/^jr) and its first n derivatives are continuous in {a x , #„), show that 

/(x) = (^_a 1 ). ..(x-anY-^, 

where £ is some number between the greatest and the least of the numbers 
x, a x , . , . , <*„. 



2. In particular, if a x — a 2 = . . . = a n = a, then 



/w =^/.«), 



where | lies between .* and a. 



CHAPTER VI. 
ON THE EXPANSION OF FUNCTIONS. 

69. The Power-Series. — To expand a proposed function, in 
general, means to express its value in terms of a series of given func- 
tions. This series has, in general, an infinite number of terms, and 
when so must be convergent. 

We confine our attention here to the expansion of a proposed 
function in a series of positive integral powers of the variable, based 
on the Theorem of Mean Value. 

The problem of the expansion of a proposed function in an 
infinite series of positive integral powers of the variable does not 
admit of complete solution in general, when we are restricted to real 
values of the variable, for the reason that the values of the variable 
at which the function becomes infinite enter into the problem, 
whether these values of the variable be real or imaginary. In the 
present chapter we shall confine the attention to those simple func- 
tions whose expansions can be readily demonstrated in real variables, 
relegating to the Appendix * a more complete discussion of the gen- 
eral problem. 

70. Taylor's Series. — If in the formula of the Theorem of Mean 
Value, 

n 

f(x) =£ (x - a) y «{a) + R„ , (I) 

r-o 

the derivatives f r (a), r — 1, 2, . . . , at a, are such that the series 

r = o 

has a finite limit when n = 00 , and we also have 

£ *n = O, 

n= 00 

then for the values of x and a involved we have 

A*) = A") + C* - «)/*(«) + —r^-/"^) + • • • (T) 



This is called Taylor s formula or series. 



* See Appendix, Notes 6, 7, 8. 

82 



Art. 7 i.] ON THE EXPANSION OF FUNCTIONS. 



■ s 3 



We may use any of the different forms of R n we choose in show- 
ing £R n = o. 

71. Maclaurin's Series.— Under the same conditions as in § 70 
if a = o, 

A*) =/(°) + ■*/» + ~/"(o) + . . . ( M ) 

This is called Maclaurin's formula* or series. 

The series (M) generally admits calculation more readily than 
does Taylor's (T), because usually the derivatives at o are of simpler 
form than those at an arbitrarily selected value of the variable a. 

EXAMPLES. 

1. Any rational integral function or polynomial /(.*•) can always be expressed as 
/(a) -f (x - a)f\a) + . . . + ( * ~ ** /»(«), 
where n is the degree of the polynomial/^). 

For, since/ is of the nth degree, all derivatives of order higher than /« are o. 
Consequently the theorem of mean value gives 

r = o 

whatever values be assigned to x and a. 

In particular, we may put a = o, and have 

/(*) = /(o) + xf{o) + . . . + ~f-{o\ 

and this must be the polynomial considered when arranged according to the 
powers of x. 

2. We may define as a transcendental integral function one such that all of its 
derivatives remain determinate and non-infinite for any assigned value of the 
variable. 

Any such function can be calculated by either Taylor's or Maclaurin's series 
for any finite value of the variable, whatever. 

For if / be such a function, then, whatever be the assigned number a, we have 
' ( x — a) n -ri 



£ 



since / M+I (?) is finite for any £■ between x and a, for all values of n. Also. 
(x — a) n +*/(n -\- 1)! has the limit o when n = 00 (see § 15, Ex. 9). 

Moreover, the series is absolutely convergent (Introd., § 15, Ex. 10), since 

r=o r=o 

where Mis a finite absolute number not less than the absolute value of any deriva- 
tive of/ at a. The series on the right is absolutely convergent, since 

V - a\ 



£' 



n = co 

see § 15, Ex. 10. 



!M]<\i, 



* This formula is really due to Stirling. 



84 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI. 

Therefore, if f(x) be any transcendental integral function, we have for any 
assigned value of x or a 

f{x) =/(«) + (x - «)/'(«) + K 2SL*tf\d) -f- • • • 



Also, 



Ax) =/(0) + x/'(0) + ^y/"(0) + 



Such functions are sin x, cos x, e x . 

3. Show that if f{x) is any transcendental integral function as defined in Ex. 2, 
then f(#x -{- q) can be expanded in Taylor's series for any assigned values of/, q, 
x and a. 

This follows immediately from 2, since 

\db?A* x + q) = P"f n U x + 9)' 

4. To expand e x by Maclaurin's formula. 

We have D r e x = e x for all values of r. At o we have 
D r e* — e° = 1. Also, 



£x 



J = o. 



(» + 1) 

Hence, substituting in Maclaurin's series, we have 

x l x' A 
e* = I + * + -j + --, + ■ • 



r= o 

In particular, when x = I, 

,= . + 1 + ^+1 + ^, + ..., 

which gives a simple and easy method of computing e to any degree of approxima- 
tion we choose. 

5. To compute sin x, given x, by Maclaurin's formula. 

sin = 0, D 2 "- 1 sin o = (— I)"- 1 , and D 2n sin = 0, 

by Ex. 5, § 56. Therefore 

x 3 x 5 x 1 , 

6. To compute in the same way cos x, given x. 

By Ex. 5, § 56, cos 0=1, Z* 2 "- 1 cos — 0, Z> 2 « cos o = (— i)». 

r* 2 x 4 x 6 
... cos*=i-- + _-- + ... 

The derivatives of sin x and cos x being always finite, these functions are trans- 
cendental integral functions and it is unnecessary to examine the terminal term R n . 
The limit of R H , however, is very readily seen to beo, since we have respectively 



x M +i / n \ 

*•= (^+i)i ain («+?*> for 

*«+' I. n \ 

— - — ■ -cos £-f--7r , for 

(« -4- ij! v 2 y 



Art. 71. J ON THE EXPANSION OF FUNCTIONS. 85 

7. The binomial formula for any real exponent. 

Consider the expansion of (1 -\- x) a by Maclaurin's series, when a is any 
assigned real number. 
We have 

D>'[1 -f x)* = a (a — 1) . . . (a - n -f i)(i -f x)*-". 
.: [D*(i + x)*] x=0 z=a(a - 1) . . . (a - n + l). 
Substituting in Maclaurin's series, we have 

ala — I) a(a — i)(a — 2) 
1 + ax + ,, V + J £ ' *" + ' ' ' 

The quotient of convergency, § 15, Ex. 9, of this series is 

|X|. (I) 



/\a - n 
I^TT- 



Therefore the series is absolutely convergent when \x\ < 1, or for all values of x 
in ) _ It _|_ i(. For !x| > 1, the series is 00 . 
Also, by (C), §.65, or § 68, (2), 

(*-€)» «(«-!)..■(«-») 
^«-* ■»!.-■ (l + |)«+x-a • ( 2) 

Whatever be the value of | between x and o, so long as \x\ < 1 we have 
*a — n x — £ 



f 



For this limit is the same as 

"l-r-g 



/ 



which is less than 1 when o<x<i. If jt < o, put x = — x' and § = — §' 
Then the limit is equal to 

- w - r 



/ 



1 - r 



But jr' — £' < 1 — §', since q < *'- < I and o < £' < *'. 

Inequality (3) being true, £R n = o, in (2). Therefore the series is equal to the 
function for the same values of x for which the series is absolutely convergent. 

, a(a — 1) „ , a(a — l)(a — 2) , 

... (I + x) a = ! + ax + * g| V + -^ ^ k« + . . . 

for all values of x in )— 1, -J- i( , and the equality does not exist for any value of 
x for which \x\ > I. 

8. Expand log (1 -f- x) by Maclaurin's series. 
Let fix) = log (1 + x). 

and /*(o) = (- 1 )«+i(« -1)!. 

Substituting in Maclaurin's series, we get 



The convergency quotient of this is 

n 



£ 



-x\ = \x\. 



S6 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI. 

The series is therefore absolutely convergent for \x\ < I, and is oo for \x\ > I. 
Also, we have, by (C), § 65, 

Whatever may be £ between x and o when |jr| < 1, we have, as in Ex. 7, 

x— £ 1 



/ 



< I. 



Therefore £Rn = o, and 

log (I + x) = x - $x* + $x* - ix* + . . . (I) 

This series converges too slowly for convenience, that is, too many terms have 
to be calculated to get a close approximation to the value of log (I -f- x). 
By changing the sign of x, 

log (1 - x) = - X - .&* - & - . . . (2) 

By subtracting (2) from (1), 

log L±^ = 2{x + **• + |*» + . . .) (3) 

If w and w are any positive numbers, put 

I + x n 4- m m 

— ! — = , then 



2n -\- m 



Substituting in (3), 



I n -\- m \ _ t m ^_\_ nf> 1 m b \ 

° g \^~) ~ 2 \^n~+^>i ' 3 &n + ™? + 5 {an + mf + ' ' 'J' 

a series which converges rapidly when « > w, and gives the logarithm of w -f « 
when log w is known. 

The logarithms thus computed are of course calculated to the base e. To find 
the logarithm to any other base, we have 

logaj =,— • 

log, a 

72. Observations on the Expansion of Functions by Taylor's 
Series. — The expansion of a given function by the law of the mean 
is rendered difficult, in general, because of the complicated character 
of the flth derivative which it is necessary to know in order to get the 
law of the series and test of its convergency. 

Still more difficult is the investigation of the limit of R n . This 
latter investigation is usually more troublesome than the question of 
convergency of the series because of the uncertainty regarding the 
value of the number £. The only information we have with regard 
to £> is that it is some number which lies between two given num- 
bers. Moreover, we know that £ is a function of n and in general 
changes its value with n. It is therefore necessary that we should 
show that jQR„ = o for all values of <<; between x and a, in order to 
be sure that £R n is o for the partici^ar value £ involved in the Inw 
of the mean whatever may be that number t c ; between x and a. In 
the deduction of the form R n in the Integral Calculus, Chapter XIX, 



Art. 73-] ON THE EXPANSION OF FUNCTIONS. 87 

§ 152, it is there shown that not only is it sufficient that we should 
consider all values of t, in the interval (a, x), but it is also necessary. 
The equality of the function and the series depends on R n vanishing 
for all values of B in (a, x)* 

It is desirable therefore, that we should have such general laws 
with regard to the expansion of functions as will enable us, as far as 
it is possible, to avoid the formation of the ;/th derivative and the 
investigation of the remainder term R n , and which will permit us to 
state for certain classes of functions determined by general properties 
that the equivalence of Taylor's or Maclaurin's series with the func- 
tion is true for a certain definite interval of the variable. The 
general discussion of this subject is too extensive for this course. 
We give in the next article some observations which will be of assist- 
ance in simplifying the problem. In the Appendix a more general 
treatment of the question is discussed. 

73. Consider a function f{x) and its derivative f'(x). We can 
state certain relations between a primitive and its derivative, with 
regard to the corresponding power series as follows: 

Cauchy's form of the law of the mean value applied to each of 
the functions f{x) and/" 7 ^) gives 

/(•v) =A«) + (* - *>/'(«) + • ■ • + ^—/"i") + *„, ( 1 ) 



/»=/» + (.v - a)/"{a) + ...+ { * a }"f "(a)+IK , (2) 



where 



(«-i)! 



(x - £)* 



R n = {x _ a) v _,'/ »"(£), (3) 



71 



R - = (* - ")^^yr/* +1 (£')- (4) 

I. We observe that the quotients of convergency of (i) and (2), 
as obtained by taking the limit of the quotient of the {n -j- i)th 
term to the «th term, have the same value, for 

/x -a f n+1 (a) £ x — a f n +\a) 

r x-af'»(a) t 1 W 

X » + ' /"(") [ n + i) ■ 

« = oo 

* In the theorem of the mean, (I), § 70, the series 
" (x — «)" 



-fn {a ) 



may be absolutely convergent and yet not equal to the function f(x). For Prings- 
heim's example, see Appendix, Note 8. 



PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI. 
Therefore, if 



/ 



n+^L\ = R (5) 

is a finite determinate number, then the two series 

/(a) + {x - a)f\a) + fc=^)_/>) + . . . 
and 

/'(a) + (* - a)f"(a) + <£Z^l/'"( a ) + . . . 

are absolutely convergent in the common interval )a — R, a -f- R( , 
and are both oo for any value of x outside of this interval. 

The number a is called the base of the expansion, or the centre 
of the interval of convergence. The number R is called the radius 
of convergence. 

II. We observe that if, for all values of B, between x and a, we 
have 

■*' w <<i., m 



I 



/"(£) 



in which B, has the same value wherever it occurs, then, § 15, Ex. 9, 
must (3) and (4) be o when n = 00 whatever be the value of B, 
between x and a in (3) or (4). 

Consequently, if we have determined (5) for any function and 
shown that (6) is true for values of x in the interval of convergence, 
then this function, its derivative or its primitive is equal to the 
corresponding Taylor's series in the common interval 

)a — R, a + R(. 

EXAMPLES. 

1. Having proved that the requirements in § 73 are satisfied for (1 -f- x) a , and 
this function is equal to its Maclaurin's series for all values of x between — 1 and 
-j- 1. and for no values of x outside these limits, it follows immediately, in virtue 
of § 73, that log (1 -\- x) is equal to its Maclaurin's series in the same interval, 
since 

D log (1 + x) = (1 + *)-i. 

2. The function tan - l x is equal to its Maclaurin's series for x 2 < 1. For 

D tan-i.r = -, 

I + .A-'"" 

and v- < I is the interval of equivalence of (1 -}- Jf8 . ,— l with its Maclaurin's series. 
M< ireover, since 

(, +**)-I =_- I _** + **-.*« + 

and the primitive of (I -f- x 2 )— ' is tan - *x, and tan- J o r= o, we have, by §73, 

tan-'* = x — fr 5 + fx- 5 - |x 7 + . . . , 
for - I < x < + I. 



Art. 74-] ON THE EXPANSION OF FUNCTIONS. 89 

We can verify this result directly, for 

In — 1)! 
D» tan-i.v = (- I)*-* -i '— sin (n tan-ijr-x). 

.-. [Z>" tan-ia-] = (- l)*-i(» — 1) ! sin(^«7T). 

Also, sin 2« —J = o, sin (2t?i -4- 1) — = (~ i)«. 

Therefore the Maclaurin's series for tan— ^ is 

x - \x* + Ax* - . . . , 

which has the interval of absolute convergence )— 1, -f- i(. 
For R n , in Lagrange's form, we have 

x n sin In tan- 1 :*: -1 ) 
Rn — 5 ^_± 

n (I + ?)i» 

the limit of which, for « = oo , is o when \x\ < 1. 
In particular, if x — tan \it — 1/^3, then 

2V3 33 5 .5 7j 

which can be used to compute the number 7t. A better method, however, is given 
below 

3. For all values of \x\ < I we have shown that 

(1 - **)-* = 1 + K + — ** + ^1 «•+... 
' \ i \ 2# 4 2.4.6 ' 

But a primitive of (1 — x 2 )—* is sin— *x, and since sin— : o = o, we have, by 
§73, 

sin- 1 * = x4--^4— -^4 . . . 
1 2 3 T 2-4S 

for x in )— I, -f- !(• 

In particular, since \it — sin -1 -£, we have 

6 2'2-3 2 3 "'2.4.5 2 5 ~ r ' 

from which 7T can be computed rapidly. 

4. Determine the Maclaurin's series for cos— x jt, cot -1 .*-, sec— l x, esc— r x. In 
each case determine the interval for which the function is equal to the series. 

74. We can find the nth. derivative of sin _I x without difficulty, 
but it would be difficult to evaluate the corresponding limit of R n by 
the direct processes of Maclaurin's formula. 

Observe that the coefficients in the power series for sin -I „r can be 
determined from Ex. 38, § 58, where we have 
(1 — x 2 )D n+2 sin-^v — (2tt -4- i)xD H+1 sin -1 * — n 2 D n s\nr x x — o. 
.-. D n+2 sjn _1 o = n 2 D n sin _I o. 

When we have found D sin _I o, D 2 sin _I o, the other derivatives at 
o can be found directly, and the interval of the convergence of the 
series established. The interval of equivalence of the function and 
the series by evaluating £R n is a matter of considerable difficulty. 



90 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI. 

In the text we go no further into this matter of the expansion of 
functions by Taylor's formula. We have made use of it to show how 
the tables of the ordinary functions and of logarithms can be com- 
puted, and the numbers e and n evaluated. 

We add a few exercises in the application of the formula. The 
cases in which the remainder term R n is inserted are those for which 
we have not established either the convergence of the infinite series or 
its equivalence with the function ; they may be regarded as exercises 
in differentiation or as applications of the Law of Mean Value. Some 
of these results will be useful later in the evaluation of indeterminate 
forms and approximate calculations. 

We observe that for the purpose of approximate calculations, if M 
be the greatest and m the least absolute value of the (n -f- i)th deriv- 
ative in the interval (a, x), the error committed in taking 

m = y ^->(«) 



lies in absolute value between 

— '—m and ] ~ ~^M, 

(«+i)! («+•!)! 

by Lagrange's form of R n . When we know the nth. derivative of the 
function to be calculated, we can thus determine beforehand how many 
terms of the series will have to be taken in order that the error shall 
not exceed a given number. 

EXERCISES. 

1. If c is the chord of a circular arc «, and b the chord of half the arc, show- 
that the error in taking 

a = 1 -(Sd-c) 

is less than — =- where a < radius of the arc. 
7680' 

2. If d is the distance between the middle points of the chord c and arc a, in 
Ex. 1, show that the error in taking 

_ 8 d* 

~ a ~ J V 

32 d* 
is less than — — . 
3 * 

3. The series 1 -\~ x -\-'x 7 -\- ... is convergent for \x\ < 1. It is infinite 
when x ^ I, and also 00 when x < — 1. Show that we can make .r converge 
to — 1 in such a way as to make the sum of the series equal to any assigned 
number we choose. 

Let x = 1, where a is any assigned number. Then we have for 

n -f- 1 

the sum of n -f- 1 terms of the series 



Art. 74. J OX THE EXPANSION OF FUNCTIONS. 91 

I — x a 

If « = 2w or 2;« -j- 1. and m = zc , this sum is respectively equal to 
1(1 + *-«) or i(i - *-*), 

one or the other of which can be made equal to any given number by properly 
assigning a. 
Show that 

4. tan .t- = x -f- \x* 4- fgX? -4- R v 

5. sec * = 1 4- §x* 4- &** + fax* 4- tf 8 . 

6. log (1 4- sin x) =x - %x* 4- ^ _ ^ + ^ 

7. ** sec x = 1 4- * 4- x* 4- f ** 4- *** 4- -V" + ^ 6 - 

8. Show that for |jt| < I we have 

/ / ;\ 1 ^ 1 • ^ x 5 

& V ^ 2 3 2.4 5 

Hint. D fcg (x 4- |/rp) = (14- ^)-l 

9. Expand sin— 1 and tan -1 — , in powers of x, determin- 

1 + -*' 2 \/l - x- 

ing the intervals of equivalence. §§ 72, 75. 

10. Expand x\/x- -\- a 2 -\- a 2 log (x -j- ^x 2 -\- a 2 ), in powers of x and deter- 
mine the interval of equivalence. 

Hint. The derivative is 2 4/a 2 4- x" 1 . 

11. Expand in like manner 

1 , 1 4- x \/~2~\- x 2 , 1 x Jz 

_ log — L L__E j _ tan-i ▼ 

44/2 I — -r 4/2 -j- x 2 2 |/2 I — * 

by using its derivative (1 4- x*)-*. 

12. Show that the nth derivative of (x 2 -f 6x 4- 8)-* at o is 



2K+2I 2"+iJ 



i)*»! -II — 



Expand the function in integral powers of x and determine the interval of 
equivalence. 

13. Show by Maclaurin's formula that 

1 

(1 +x)*=e{l -** + &*■- A**}+* 4 . 

- , , logfi4-jr) 

Hint. If j = (1 + x)*, then log/ = — 2 — '. 

.: y = e*(*\ 0{x) = I - \x + \x 2 - \x* 4- . . . , 
and the first few derivatives can be found. 

14. Compute the following numbers to six decimal places: e, it, log 2, log e 10, 
sin io°. 



CHAPTER VII. 

ON UNDETERMINED FORMS. 

75. When u and v are functions of x, they are also functions of 
each other. If, when x(=.)a, we have ti(=z)o and z>(=)o, the quotient 

u 

v 

will in general have a determinate limit when x( = )a. This limit 
will depend on the law of connectivity between u and v. The evalua- 
tion of the derivative is but a particular and simple case of the 
evaluation of the limit of the quotient of two functions which have 
a common root as the variable converges to that root. For, in the 
derivative, we are evaluating the limit of the quotient 

x — a 
when f(x) — /(a)( — )o and x — a\ = )o. 

The evaluation of the quotient u/v when x converges to the 
common root a of u and v, is but a generalization of the idea 
involved in the evaluation of the derivative. For, let <fi(x) and ip(x) 
be two functions which vanish when x = a, or, as we say, have a 
common root a. Then 







<p(a) = 


and 


f(a) = 0. 


We w 


ish to < 


evaluate the limit of the quotient 


when x(- 
Since 


0(a) 


= o, ,l(a) = 


<P(x) 

= 0, we have 








<P(x) 
f(x) 


-0(a) 








X - 


-0W 




- (7 




i-{x) - 


" *(«)" 



A" — (7 

Consequently if 0(a) and //'(-v) are differentiate functions at a, 



Art. 76.] OX UNDETERMINED FORMS. 93 

and the member on the left has a determinate limit when x( = )a, we 
have 

f<P(x) <fi'(a) 



For example, 






It may happen that a is a common root of (f>'(x) and i//(x), then 
0'(tf) = o and ^'(tf) = o. In this case we shall require a further 
investigation in order to evaluate the quotient (p/tp. For this pur- 
pose we require the following theorems: 

76. A Theorem due to Cauchy. — Let cp(x) and ip(x) be two 
functions which vanish at a, as also do their first n derivatives, but 
the {n + i)th derivatives of both <p(x) and tp(x) do not vanish at a. 
Then we shall have 

fW+'tf) = tf>(*)0 M+I (£), 

where B, is some number between x and a. 

Let z be a variable in the interval determined by the two fixed 
numbers x and a. Then the function 

/(*) = 0(*) *(*) - «*) 0(*)> 

= o when z =. a, also when 2 = ^. 

By the law of the mean, § 62, J'{z) = o for some number g = £ 
between jr and «. But, in virtue of the fact that / (#) = i/>'(a) = o, 
we have /'(#) = o. Consequently J"(z) = o for some number B, 2 
between B, x and a. 

In like manner J"\z) = o for z = B> % between Bi and a, and so 
on until finally we have 

/H-i(£) = 0«+'(£) 0(*) - #**(£) 0(*) = O, 

where B, is some number between x and #. 

If tp n+1 (z) is not o between x and a, we can divide by it. Hence 

*(«) f**(«) • 

This theorem is of great generality and usefulness. 
For example, the functions (x — a) n +*/(n-\- i)\ and 

J\x)^f{x)- ^LZ^Lfr^) 

r= o 
are such that they and their first n derivatives vanish at x = a, while the (n -f- i,th 
derivative of the first function is 1. Therefore, by the theorem just proved, we 
have 

(x — a) H + l 

which is Lagrange's formula for the law of the mean. 



94 rRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. 

This theorem can be utilized for finding many of the different forms of the 
remainder in the law of the mean. It has, however, its chief application in : 

77. The Theorem of l'Hdpital. — If <p(x) and ip(x) are two func- 
tions which vanish at a, as also do their first n derivatives, then we 
shall have 

/>(■*) r 0" +i (g) _ r ,+, (-y) 
In*) X r + \t) X *■*(*)' 

For, by Cauchy's theorem, § 76, 

0(*) = 0* + '(£) 

where B, lies between .a? and a. Hence, since B, and x converge to 
a together, we have for x(=)a 

f 0(a) 0"+-(a) 
2 *(«) r + '(*)' 
Moreover, Cauchy's theorem shows that the quotients 

<P( X ) < s 

~^y (r=i, 2 ,...,*) 

all have this same limit. 

Therefore, to find the value of the undetermined form, we 
evaluate successively the quotients of the successive derivatives until 
we arrive at a quotient no longer indeterminate. 

EXAMPLES. 

1. Evaluate, when*(=)i, the quotient 



x 1 — I 
x 2 — 3-r -f- 2 = o, when x = I. 
D(x 2 — 3.r -f- 2) = 2x — 3, = — 1. when * = 1. 
x 2 — 1 = o, when .v = 1. 
Z>(.r 2 — 1) = 2x, = 2, when x = 

- 3_ _l_ 
2JC 2 



/jr 2 — 3-r 4- 2 _ /Vv 



2. Show that 






Art. 78.] ON UNDETERMINED FORMS. 95 

3. Evaluate, when .v( = )o, the following: 



/t' x — e~ x I x sin x 
: = 2 ; / . — = o. 
sin x / .i — 2 sin x 

( = )o *(=)o 

r x( = )o, we have 

f t*-*-*-™ = 2 . / > + .-*- _* = 2 . 

/ x — sin x I vers x 

e, when x(=z)o, 

/x — sin-'x _ _ I £ (i x — b x _ lo a Aa n .r — x _ 

"sin 3 x ~ 6 ' ^J a- g * ' ^J x - sin # 



-r( = )o *(=)o 

4. Show that for x( = )o, we have 



5. Evaluate, when x(=)o, 



6. Find the limits, when x( = )o. 



/x — sin * _ I /" sin 3^ _ _ 3 /" 

^3 — 6 ' 7" x — I sin 2x 2 ' J^ 1 



cos mx ;«' 

78. The Illusory Forms. — When u and v are two functions of x, 
which are such that the functions 

u/v, uv, u — V, u v , 

tend to take any of the forms 

0/0, 00 /oo , o X CO , CO — 00 , o°, 00 °, I*, 
as x converges to a ; then when these functions have determinate 
limits for x( = )a, the theorem of l'Hopital will evaluate these limits. 

All these forms can be reduced to the evaluation of the first, 0/0, 
as follows : 

(1). 00 /oo and o X 00 reduce directly to 0/0. 

For, if u a = 00 , v a = 00 , then 

^ _ 00_ _ I/Z^ __ O 
Va ~~ «> 1 / U a O' 

and we evaluate 

I/O* 

i/u x ' 
If « a = o, v a = co , then 

1/V a o 

and we evaluate 



i/v x 
(2). In like manner, if u a = 00 , # a = 00 , then 

v a \ 1 — vju a o 



«„ — »„ = «„ I — 



provided £{v x /u x ) = 1, otherwise this form has no determinate finite 
limit and is 00 . 



96 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. 

This illusory form can also be reduced to the evaluation of the 
form o/o when x( = )a, thus : 

e~ v 

which takes the form o/o when x = a. Therefore, if £e u ~ v = e c , 
£{u-v) =c, for x( = )a. 

(3). The last three forms, o°, co °, i°°, arise from the function 
u v , which can be reduced to 0/0, thus : 

Since u = e l0 ^ u , 

.-. u v — e vloeu . 

In each of the cases o°, co °, i°°, the function v log u takes the 
form o X 00 , which can be turned directly into 0/0 and evaluated as 
in (1). 

Examples of 00 /oo and 0/00 . 

The evaluation of u/v, when u = 00 , v = 00 . for x = a, is carried out in the 
same way as for 0/0. For we have 

/0(*)_ £ */*{* ) _ f -ft*)/W*)7 

when x(=)a. If now <p{x) / f ip(x) has a determinate limit A 5^ o, when x(=)a f 
then 

X <t>w 

Therefore, for x( = )a, when <p(x) = 00 , tj)(x) = 00 , 



=£\ 






if (p'{a)/x})'(a) is determinate. 

jf /'sec x 

tan.* - ^J tan x 

Vtan x\ 2 

* = I , when #(=)$«. 



/*tan x _ /* sec 2 
r sec x ~~ T sec x ti 

x WW 

~ rx. • tai1 ~ V 

Or immediately, by Trigonometry, = sin x. 

2. Show that 



£?■-£'-■ 



when n is a positive integer. Also when n is not an integer. 
3. Show that £ x w (log jr)« = o. 



Art. 79. J ON UNDETERMINED FORMS. 97 



/I 

4. Show that + 


og(9- in) _ q 
tan 




5. Evaluate, when 


*(=)**, 




tan x 


log tan 2x 
log tan x ' 


I — sin x -f- cos x m 


tan 3-r 


sin x -f- cos .r — 1 ' 


log sin x 


sec x 


tan x 



(7C — 2u/ 2 ' sec 3.V tan 5* 

6. Show that £ (1 — x) tan \{itx) = -. 

Examples of 00 — 00 . 

7. ^(secjr — tan jt) = o. for x( = )l7C. 

8. £{x~ l — cot .r) = o, for x( = )o. 

9. •* tan .r — \it sec x{=) — I, when jr( = )^w. 
._ .r — sinjr , x I , . , 

10. 3 — ( = )6"' when *( = ) a 

11. {a*— 1 )/*(=) loga, when x(=)o. 

Examples of #». 



I 
12. (1 +jr)*(=y, 


x(=)o. 


13. (1 + **H=)i, 


x(=)o. 


14. (^+i)*(=V, 


x = 00 


15. (COS 2JCp( = )r-2, 


*( = )o. 


16. ^" == ^(=)^s 


*(=)!. 



79. General Observations on Illusory Forms. — In evaluating 
illusory forms, we may at any stage of the process suppress any com- 
mon factors in the numerator and denominator, and evaluate indepen- 
dently any factor which has a determinate limit. We can frequently 
make use of algebraic and trigonometric transformations which will 
simplify and sometimes permit the evaluation without use of the 
Calculus. 

In illustration consider the limit of 



(x-i) l ° e * inirx , when *(=)i. 

This takes the form o°. To evaluate, equate the function to_>/and 

take the logarithm, 

log (x — 1) 

... \ ogy=a -^-. '. 

log sin 71 x 

1 

/log (x — 1) f x — r 1 /'sin 7tx 
-r-— . - = f = - / sec nx. 
log sin nx Jb n cos nx n J^ x — 1 

sin nx 



98 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. 

But ;£sec rex = — i, and 

/sin7r^"_ Ctt cos tcx 
* — i ~ X ~~~ i - 

• '• £ log ^ = ] og ^J' = «• 
Hence ; £j/ = e a , when #(=)i. 

Frequently the evaluation can be simplified by substituting for the 
functions involved their values in terms of the law of the mean. 
For example, evaluate for x( — )o, 

i 

(i + *Y- e 



Differentiating numerator and denominator, we have 
x- (i + or) log (i + x) 



(i + x)- 



x\i+x) 



jQ(i _|_ x) x = e, and the limit of the other factor is, by the ordinary 
process, readily found to be — £. Hence the limit is — e/2. 

Otherwise, put for (1 -j- x) x i\& value, Ex 13, Chap. VI, 
x 11 « 7 ^ ) 

2 ■ 24 16 ' * J 

and the result appears immediately without differentiation. 



Geometrical Illustrations. 

(1). If/(«) = o, 0(a) = o, f'(a) ?* o, <p'(a) ^ o, consider the curves rep- 
resenting/ = /(x), y = <p(x). 



y=f(x) 




Fig. 1 



These curves cross Ox at x =.-. a at angles whose tangents are equal to /'(a), 
(t>'{a), or 

/'(a) = tan B v 0\a) = tan 6 2 . 

tan X 






tan 



The limit of the quotient/ (x)/0(x) is represented by the quotient of the slopes 
of these curves at their common point of intersection with Ox. 



Art. 79.] 



OX UNDETERMINED FORMS. 



99 



(2). Consider the functions x and y in 

Differentiate with respect to .r and solve for Dy. 

2JT 3 -\- 2xy 2 - 



- Dv = 



Dy takes the form 0/0 when x _ 
this, differentiate the numerator and denominator with respect to x 



2x' z y -j- 2y' A -J- d l y ' 
o, for then also y — o by (i). 



- £ Dy 






21/ 



\xy Dy 



+ (2^2 _|_ 6 y2 + d i^ Dy » 



(i) 



To evaluate 



■•• {£&)'?= I, or £Z* = ± 1. 
This means that the curve whose equation is (i) in Cartesian coordinates has 
two branches passing through the origin x = o, y — o, which is a singular point. 
There the slopes of the two branches to Ox are -f- 1 and — 1. The curve is the 
lemniscate. 




Fig. 12. 

We can find Dy at x = o. y = o for the curve (i), without indetermination by 
differentiating the equation (i) twice with respect to x. Thus 

(2a 2 - I2x 2 - 4j 2 ) = i6xy Dy-\-(vc i +i2y 1 + 2a 1 )(Dyy-\-($x' i yJ r 4y* +2a 2 y)D 2 y, 

which gives, as before, Dy =. ± 1, when x = o, y — o. 

(3). We know from trigonometry, that the radius p of the circle circumscrib- 
ing a triangle ABC with sides a, l>, c having area S, is 

abc 

Also, from Analytical Geometry, we have 

2S = I x, y, 1 
?i> .Ti> x 

I ^2' ^2' I 

where jc, >'; x l . y x ' } x v y 2 , are the coordinates of the corners of ABC. Show that 
if A, B, C are three points on a curve y=f(x), then the radius of the circle through 
these three points, when x l ( = )x, x. 2 ( — )x, is 

[l + (Dyff 



We have 



D 2 y 



L.cfC. 



' 2 = K-*) 2 + (-n-/) 2 , 

<$* = (*, _ xf + (^ - y)*, 
a*= (x 2 - Xl) 2+(y 2 -y x f. 



ioo PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. 

Also, 

x y I = xy x —yx x + y 2 (x x - x) - x 2 (y x - y). 
x x y x i 

Substitute these values in the expression for p. Observe, when x x ( = )x, p is of 
the form o/o. Divide the numerator and denominator by x x — x and let x^zz: )x. 




Fig. 



To evaluate 



xy x 



13- 



for x x ( = )x, differentiate the numerator and denominator with respect toXj and then 
let x 1 (=)x. 

"xy x —yx x 



£ 



£(xDy x -y) = xDy-y. 



x 1 ( = )x 

Therefore, when B{=.)A, 

2 y-> 



xf 



[x 2 - x) 



[I + (Z*)*]* 



y - (x 2 - x)Dy { \x 2 — x) 

The first factor takes the form o/o when x 2 = x. To evaluate it, differentiate 
the numerator and denominator with respect to x 2t and we have 

I 2(x 2 — x) _ 
1 Dy 2 - Df 

this is again o/o when x 2 — x. To find its limit when x 2 ( = )x. differentiate the 
numerator and denominator with respect to x 2 , and there results 

which has the limit i/D 2 y when x 2 ( = )x. 

Therefore when the points B and C converge to A along the curve, the circle 
ABC converges to a fixed circle passing through A which has the radius 

\ _ , (dyy ) l 



R = 






-Ml 

d'-y 
dx> 



T liis circle is called the circle of curvature of the curve jy = f(x) at the point 
x, v. and A* is called the radius of curvature. Observe that when x,(=).i and 
x., ?£ x, the circle and curve have a common tangent at A, or, as we say, are 
tangent at A. When this is the case the curve and circle both lie on the same side 
of the tangent at A. Also the circle lies on the same side of the curve in the neigh- 



Art. 7.).] ON UNDETERMINED FORMS. 101 

borhood of A. But when also x 3 (=)x the circle crosses over the curve at ./. 
The circle of curvature is said to cut a curve in three coincident points at the point 
of contact, in the same sense that a tangent straight line to a curve is said to cut 
the curve in two coincident points at the point of contact. Remembering that all 
points in the same neighborhood are consecutive, the above statement has definite 
meaning. 

Much shorter ways of finding the expression for the radius of curvature will be 
given hereafter, but none more instructive. 

EXERCISES. 

1. Evaluate, when x(=)o, 

sin 2 jc — 2 sin x 



/ e x _ 2cos x -\- e~ x _ P sin 2x -f- 2 1 

x sin x T cos x 

2. Also, for the same limit of x, 

f sin 4x cot x /'sin i-x cos 2x 



= 4- 



/ sin 4-r cot x _ /'sin ±x 

vers 2x cot 2 2x ' T vers x 

en x(=)o, 

r m sin x — sin mx m P ta 

J x(cos x — cos mx) 3 ' I n 



cot X 



3. Show, when x(=)o 



tan «jr — « tan x 



sin x — sin nx 



— 2. 



4. If x{ = )o, then 



/(x — 2)ex + x + 2 I r TtX 2 
T~Z —vT~ — = -z J ./ (1 - ^) tan — = -. 
{e x — if T 2 % 

/ a \* I a\^ I aV°e* 
{ C05 x) ' ( COS x) ' ( C0S ^j ' 



*( = ) 
5. Evaluate for x = 00 , 



6. Find the limits, when Jt(=)o, of 



tan a: / I \ sin x 

, ( — J , (sin x) sin * (sin .r) tan * 



7. Find the radius of curvature of the parabola y 2 = \px at any point x, y, and 
show that at the origin it is equal to 2p. 

8. Evaluate 

/ ( fl 2 _ 02y* -f (a -. Q)8 _ |/i^ 

( fl »_e3)i +(a _ e) i~ 1 + ^4/3"' 



#sin * ^ 

9. -= : — —( = )a log a. when x(=z)iie, 

log sin .* 

V_ 4 + ^-f 2 cos x _ I 

#* 6' 



*( = )0 



11. ^ jtc* = 00 






j: 2 24 



102 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. 

H £d l ax*+bx + c \ a 
' £ dx \ px + q j p 



15. Evaluate, when x = oo , 



\/x -f a — \/x -f- b, \/x 2 -j- ax — x, ax sin (cja*). 
16. Find where the quadratrix 

TCX 

y = x cot — 
y ia 

crosses the y axis. 

t /Y— « r )=f- 

2^ 2x tan ^ / 6 



*( = )o 



CHAPTER VIII. 
ON MAXIMUM AND MINIMUM. 

80. Definition. — A function f{x) is said to have a maximum 
value at x = a when the value of the function, /(«), at a is greater 
than the values of the function corresponding to all other values of 
x in the neighborhood of a. 

The function is said to have a minimum value at a when /(a) is 
less than_/~(:v) for all values of x in the neighborhood of a. 

In symbols, /(x) is a maximum or a minimum at # according as 
/(.r) -/(«) 

is negative or positive, respectively, for all values of x ^ « in 
(# — e, a -j- e) the neighborhood of a. 

81. Theorem — At a value a of the variable for which the func- 
tion f(x) is differentiate and has a maximum or a minimum, the 
derivatives^) is o. 

At a value a at which /"(**") is a maximum or a minimum, by defini- 
tion the differences 

/(x')-f(a) and f{x")-f(a), 

where x' < a < jv", have the same sign. 
Consequently the difference-quotients 

a*-)-a<) and ? „ = /k;)-/(«) 

x — a x — a 

have opposite signs for all values of x' and x" in the neighborhood 
of a, since x' — a is negative and x" — a is positive. Therefore, 
since q' and q" have the common limit /'(a) when .*:'( = )a and 
x"( — )a, we have 

= 2 1/» 1 =0. 

Hence /" '(a) = o. 

Notice that at a maximum value of the function the derivative is o, 
and since, by definition, the function must increase *up to its maxi- 
mum value and then decrease as x increases through the neighbor- 
hood of a, the derivative on the inferior side of a is positive and on 
the superior side is negative, § 60. 

Hence, at a maximum, a, the derivative, f'(a), is o and J\x) 
changes irom positive to negative as x increases through a. 

103 



104 PRINCIPLES OF THE DIFFERENTIAL CALCULI'S. [Ch. VIII. 



In like manner, at a minimum, x = a, the derivative, /'(a), is o, 
and_/"'(.v) changes from negative to positive as x increases through a. 

Conversely, whenever these conditions hold, then the function 
has a maximum or a minimum value at a, accordingly. 
For example: 

1. Let f(x) = x 2 — 2x -j- 3. 

... /(*)= 2(X- I). 

We have /'(I) = o. Also for x < 1, we have f'(x) negative, and for x > 1, 
f'{x) positive. 

Hence /(i) = 2 is a minimum value of f{x). 

2. Let f(x) = — 2X 1 -f Sx — 9. 

.-. f(x) = 4{2-x). 
We have f{z) = o, /'(2 - e) = +, /'(2 -f e) = — . 
.-. /(2) = - I is a maximum. 

82. The condition /'(a) = o is necessary, but it is not sufficient, 
in order that the function f{x) shall have a maximum or a minimum 
value at a. For the derivative f'(pc) may not change sign as x 
increases through a. It may continue positive, in which case J\x) 
continues to increase as x increases through a; or f\x) may be 
negative throughout the neighborhood of a, in which case the func- 
tion continually diminishes as x increases through a. These condi- 
tions can be illustrated geometrically thus: 



Geometrical Illustration. 

Represent j =/(x) by the curve ABCDE. Then f'(x) is represented by the 
slope of the tangent to the curve to the x-axis. At a maximum or a minimum, 
f'(x) = o or the tangent to the curve is parallel to Ox. In the neighborhood of 




Fig. 14. 



a maximum point, such as A or C, the curve lies below the tangent, and the 
ordinate there is greater than any other ordinate in its neighborhood. In like 
manner at a minimum point, such as B or D, the points fi, D are the lowest points 
in their respective neighborhoods. At a point E the tangent is parallel to <9.v, 
and f\x) = o, but the curve crosses over the tangent and is an increasing function 
at E, also the derivative f'(x) is positive for all values of x in the neighborhood. 

It will frequently be impracticable to examine the signs of the 
derivative in the neighborhood of a value of x at which f'(pc) = o. 
A more general and satisfactory investigation is required to discrimi- 
nate as to maximum and minimum at such a point. 



Art. 83.] ON MAXIMUM AM) MINIMUM. 105 

83. Study of a Function at a Value of the Variable at which 
the First n Derivatives are Zero. 

(1). Let /(a) be a function such that /'{a) ^ o. Then by the 
law of the mean, §§ 62, 64, 

f{x) -f{a) = (x - a)f\B). 

By hypothesis, /'{a) ^ o is the limit of f\x) and of /'(£) as 
.r( = )<7, since B, lies between x and a. Consequently we can always 
take x so near a that throughout the neighborhood of a we have 
_/"(£) of the same sign as f'(a) for all values of x in that neighbor- 
hood. Hence, as x increases through the neighborhood of «, the 
difference f(x) —f(a) changes sign with x — a; and by definition 
f(x) is an increasing or decreasing function at a according as /'(a) 
is positive or negative respectively. 

(2). Lety'^z) = o and f"(a) 7^ o. Then 

/(.v) -/(a) = ( - V ~ U) /"(g). 

Throughout the neighborhood of #, f'\&>) has the same sign as 
its limit /"{a) ^ o, and therefore does not change its sign as x 
increases through a. But, as {x — a) 2 also does not change sign as 
x passes through a, we have the difference 

/(•*) -A"), 

retaining the same sign for all values of x in the neighborhood of a, 
and having the same sign as /"(a). Consequently, by definition, the 
function f{x) has a maximum or a minimum value f{a) at a according 
zsf"{a) is negative ox positive respectively. 

(3). Let /'(a) = o, f"(a) = o, /'"(a) ^ o. Then 

As before, in the neighborhood of #, /'"{&>) has the same sign as 
its limit f'"{a) ^ o. But (.# — «) 8 changes its sign from — to -f- 
as x increases through a. Therefore the difference 



i .-,' 



A*) -/(") 

must change sign as x increases through a, and f{x) is an increasing 
or decreasing function at a according as f'"(d) is positive or negative. 

(4). Let /'(a) =/"(*) = . . . =/» = o, but /*«(«) ^ o. 

Then, by the law of the mean, 

/(*) -/(«) = i ^jr> f *w- 

In the neighborhood of a, / M+1 (£) has the same sign d,sf n+x (a). 
If « + 1 is odd, then (x — a) n+l and therefore f(x) —/(a) change 
sign as x increases through a; and _/*(.*:) is an increasing or decreasing 



106 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIII. 

function at a according as f n+1 (rf) is positive or negaiive. If, how- 
ever, n -\- i is even, then (x — a) n+1 does not change sign, nor does 
the difference f{x) —/{a), as x increases through a; consequently 
f{x) is a maximum or a minimum at # according as f n+l (a) is nega- 
tive or positive. Hence the following 

84. Rule for Maximum and Minimum. — To find the maxima 
and minima values of a given function f{x), solve the equation 
f\x) = 0. If a be a root of the equation f'(x) = o, and the first 
derivative of f{x) which does not vanish at a is of even order, say 
f 2n {a) ^ o, then /"(a) is a maximum \if %n {a) is negative, or a minimum 
\i/ 2n {a) is positive. 

EXAMPLES. 

1. Find the max. and min. values, if any exist, of 

(p(x) = x* — gx 2 -)- 24-r — 7. 

We have <p'(x) = 3(x 2 — 6x + 8) = 3(x - 2)(x — 4). 

... 0'{z) = o, 0'( 4 ) = o. 
Also, 0"(.r) = 6(* - 3). 

... <p"(2)=-, 0"( 4 ) = +. 

0(2) z= -|- 13 is a maximum, 0(4) = 9 a minimum. 

2. Investigate for maxima and minima values the function 

(p{x)= e x -f e~ x -f 2 cos x. 
We have 0'(o) = <p"(o) = 4>'"(o) = o, IV (°) = 4- 

0(o) = 4 is a minimum. Show that o is the only root of 0' (•*")• 

3. Investigate x 5 — 5.x 4 4- 5-r 8 — I, at # = I, x = 3. 

4. Investigate x 3 — 3-r 2 + 3 X + 7> at x = J « 

5. Investigate for max. and min. the functions 

X 3 _ 3 y2 _|_ 6x 4. 7> ^3 _ gx _|_ I5x _ 3# 

3-r 5 — 125.x 3 -|- 2l6ar, x 3 -j- 3^ 2 + 6.r — 15. 

6. Show that (I — x + -r 2 )/(i -f- x — x 2 ) is min. at x = ^. 

7. If -v;'(j — •*") = 2 # 3 > show that jy has a minimum value when x = a. 

8. If 3^ 2 j 2 -f- x>' 3 4- 4^7^ = o, show that when x — 3^/2, then y — — 3a 
is a maximum. D 2 y being then — 

9. If 2x h -f- 3<rj' 4 — x 2 ;' 3 = o, then jr = $ s a makes jj' = $*a a minimum. 

85. Observations on Maximum and Minimum. 

(1). We can frequently detect the max. or min. value of a func- 
tion by inspection, making use of the definition that there the neigh- 
boring values are greater or less than the min. or max. value 
respectively. 

For example, consider the function 

ax' 2 -{- bx -J- c. 
Substitute y — bjia for x. The function becomes 
\ac - b 2 

— z; — ^ °y > 



Art. 85.] ON MAXIMUM AND MINIMUM. 107 

which is evidently a maximum when/ = o and a is negative, and a minimum 
when j' = o and a is positive. 

(2). Labor is frequently saved by considering the behavior of the 
first derivative in the neighborhood of its roots, instead of finding the 
values of the higher derivatives there. 

For example, see Ex. 6, § 85, and also 

0{x) = (x - 4) 5 (x + 2)*. 

Here <f>\x) = 3(3^ - 2)(x - ^Y(x -f 2) 8 . 

0' passes through o, changing from -j- to — as x increases through — 2; there- 
fore 0( — 2) is a maximum. 

0' passes through o, but is always positive as x increases through 4 ; therefore 
0(4) is an increasing value of (p{x). Also 0' passes through o, changing from 
— to -I- as x increases through 2/3, and the function is a minimum there. 

(3). The work of finding maximum and minimum values is fre- 
quently simplified by observing that 

Any value of x which makes fix) a maximum or a minimum also 
makes Cf{x) a maximum or a minimum when C is a positive constant, 
and a minimum or a maximum when C is a negative constant. 

f(x) and C -\-f(x) have max. and min. values for the same values 
of x. 

(4). If n is an integer, positive or negative, f{pc) and \/{x) \ n have 
max. and min. values at the same values of the variable. In particu- 
lar, a function is a maximum or a minimum when its reciprocal is a 
minimum or a maximum respectively. 

(5). The maximum and minimum values of a continuous function 
must occur alternately. 

(6). A function J\x) may be continuous throughout an interval 
(a, /?), and have a maximum or a minimum value at x = a in the 
interval, while its derivative f\x) is 00 at a, but continuous for all 
values of {x) on either side of a. 

In this case, to determine the character of f{x) at a, we can use 
(1) or (2) as a test. Otherwise we can consider the reciprocal 
i/y , (x) f which passes through o and must change sign as x passes 
through a, for a maximum or a minimum o(f(x) at a. 

EXAMPLES. 

1. Consider <p(x) = (x - 2) 1 + 1. 

is a one-valued and continuous function and is always positive. It clearly 
has a minimum at x — 2, where <p(x) = 1. We have 
. N 2 1 
(•*■) = ~ 1 » 

3 (X - 2) 3 

and 0'(2) = 00 . Also, 0'(2 — h) is negative and 
0'(2 -f- h) is positive. 

2. In like manner 

ij){x) = 1 - {x - 2) ! 

has a maximum at x = 2. ^ IG> I 5- 




ioS PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cn. VIII 

3 

3. Consider <p{x) = I -\- (x — 2) B , 
which is also uniform and continuous. 
We have 

5 (x—2) 5 

which is -f- °° when x = 2, but is always -f- in the neigh- 
Y lG j^ borhood of 2. Therefore, at x = 2, <p(x) is an increasing 

function. 

In like manner I — (x — 2) 5 is a decreasing function at x — 2. 
(7). In problems involving more than one variable we reduce the 
conditions to a function of one variable by algebraic considerations. 
Otherwise, we can frequently make a problem involving more than 
one variable depend on one which can be solved by elementary con- 
siderations. 

For example, the sum of several numbers is constant; show that their product 
is greatest when the numbers are equal. 

First, take two numbers, and let 

x -\-y-c. 

Then \xy = (x -f- yf - (x - yf = c 2 - (x - y)\ 

which is evidently greatest when x — y. 

Let x -f- y 4. z = c . 

Then, as long as any two of x, y, z are unequal, we can increase the product 
xyz without changing the third, by the above result. Therefore xyz is greatest 
when x = y = z. The method and result is general, whatever be the number 
of variables. 

EXERCISES. 

1. Find the maximum and minimum values of y, where 

y = (x - i)(x - 2f. 

2. Find the max. and min. values of 

(i). 2X 3 — 15a- 2 -f 36* -f- 6. 

(2). (X - 2){X - 3)2. 

(3). x* - 3 * 2 + 6 * + 3- 
(4). 3-r 5 - 25.x 3 -f 6ox. 

3. Show that(x 2 -f- x -f- i)/(x 2 — x -f- 1) has 3+ 1 for max. and 3- 1 for min. 

4. Find the greatest and least values of 

a sin x -\- l> cos x and a sin 2 x -f- b cos 2 jr. 

5. Investigate (x 2 -f 2x — i$)/(jc — 5), and also 

x l — jx -\- 6 
x — 10 ' 
for maximum and minimum values. 

6. The derivative of a certain function is 

(* - l)(* - ^y 2 (x - 3 )»(* - 4)*; 
discuss the function at x ■ = I, 2, 3, 4. 

7. Find the max. and min. values of 

(a), (x - i)(x - 2){x -3), (<•). x(i - *)(i - x 2 ). 

[b). -v* - &*» + 22.r 2 - 24-r, (/)• (-V 2 - !)/(** + 3 ) 3 , 

(c). (x — a) 2 (x — t>), (g). sin x cos*or, 

[d). (x -a)\x- !>)\ (//). (log.v)/.v. 



ART. 85.] ON MAXIMUM AND MINIMUM. 109 

8. Show that the shortest distance from a given point to a given straight line is 
the perpendicular distance from the point to the straight line. 

9. Given two sides a and b of a triangle, construct the triangle of greatest 
area. 

10. Construct a triangle of greatest area, given one side and the opposite 
angle. 

11 . If an oval is a plane closed curve such that a straight line can cut it in only- 
two points, show that if the triangle of greatest area be inscribed in an oval, the 
tangents at the corners must be parallel to the opposite sides. 

12. The sum of two numbers is given; when will their product be greatest? 
The product of two numbers is given ; when will their sum be least ? 

13. Extend 12 by elementary reasoning to show that if 

2(x r ) =5 x x + . . . + x H = c, 

n 

then IL(x r ) = x x . . . x„ 

is greatest when x x = x 2 = . . . = x M . 

14. Apply 13 to show that if. -\- y -\- z = c, the maximum value of xy 2 z % is 
^/43 2 - 

15. Show that \i x -\- y -\- z z= c. the maximum value of x l y m z n is 

l l m m n n c l + m + n 



(/ 4- m + n)i+™+* 

16. Find the area of the greatest rectangle that can be inscribed in the 
ellipse. (Use the method of Ex. 15.) 

x 2 y 2 

—, + -77, = I. [Ans. 2abA 

a 1 b l J 

17. Find the greatest value of Sxyz, if 

x % y 2 z 2 I" abc 

a* + T> + 7> =1 - [_ 



3 V3 

This is the volume of the greatest rectangular parallelopiped that can be 
inscribed in the ellipsoid. 

18. Show that the greatest length intercepted by two circles on a straight line 
passing through a point of their intersection is when the line is parallel to their 
line of centres. 

19. From a point C distant c from the centre O of a given circle, a secant is 
drawn cutting the circle in A and B. Draw the secant when the area of the 
triangle A OB is the greatest. [With C as a centre and radius equal to the diagonal 
of the square on c, draw an arc cutting the parallel tangent to OC in D. Then 
DC is the required secant. Prove it.] 

20. A piece of wire is bent into a circular arc. Find the radius when the seg- 
mental area under the arc is greatest and least. \r = a /ft, r = 00 .] 

21. Find when a straight line through a fixed point P makes with two fixed 
straight lines AC, AB, a triangle of minimum area. [P bisects that side.] 

22. The product xy is constant; when is x -\- y least ? 

23. An open tank is to be constructed with a square base and vertical sides, 
and is to contain a given volume ; show that the expense of lining it with sheet lead 
will be least when the depth is one half the width. 

24. Solve 23 when the base is a regular hexagon. 



no PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIII. 

25. From a fixed points on the circumference of a circle of radius a, a perpen- 
dicular A Y is drawn to the tangent at a point P ; show that the maximum area of 
the triangle APYU 3 V 3V 2 / 8. 

26. Cut four equal squares from the corners of a given rectangle so as to con- 
struct a box of greatest content. 

27. Construct a cylindrical cup with least surface that will hold a given 

volume. 

28. Construct a cylindrical cup with given surface that will hold the greatest 
volume. 

29. Find the circular sector of given perimeter which has the greatest area. 

30. Find the sphere which placed in a conical cup full of water will displace 
the greatest amount of liquid. 

31. A rectangle is surmounted by a semicircle. Given the outside perimeter 
of the whole figure, construct it when the area is greatest. 

32. A person in a boat 4 miles from the nearest point of the beach wishes 
to reach in the shortest time a place 12 miles from that point along the shore; he 
can ride 10 miles an hour and can sail 6 miles an hour ; show that he should 
land at a point on the beach 9 miles from the place to be reached. 

33. The length of a straight line, passing through the point o, />, included be- 
tween the axes of rectangular coordinates is /. The axial intercepts of the line are 
a, ft, and it makes the angle B with Ox. Show that 

(a). I is least when tan 6 = (b/a) 5 . 

(b). a + ft is least when tan B = {b/af. 

(c). aft is least when tan B = b/a. 

34. Find what sector must be taken out of a given circle in order that the 
remainder may form the curved surface of a cone of maximum volume. 

[Angle of sector = 271(1 — V2/3).] 

35. Of all right cones having the same slant height, that one has the great- 
est volume whose semi-vertical angle is tan~i V2. 

36. The intensity of light varies inversely as the square of the distance from 
the source. Find the point in the line between two lights which receives the least 
illumination. 

37. Find the point on the line of centres between two spheres from which the 
greatest amount of spherical surface can be seen. 

38. Two points are both inside or outside a given sphere. Find the shortest 
route from one point to the other via the surface of the sphere. 

39. Find the nearest point on the parabola j 2 = 4/.V to a given point on the 
axis. 

40. The sum of the perimeters of a circle and a square is /. Show that when 
the sum of the areas is least, the side of the square is double the radius of the circle. 

41 . The sum of the surfaces of a sphere and a cube is given. Show that when 
the sum of the volumes is least, the diameter of the sphere is equal to the edge of 
the cube. 

42. Show that the right cone of greatest volume that can be inscribed in a given 
sphere is such that three times its altitude is twice the diameter of the sphere. 

Also show that this is the cone of greatest convex surface that can be inscribed 
in the sphere. 

43. Find the right cylinder of greatest volume that can be inscribed in a given 
right cone. 



ART. 85.] ON MAXIMUM AND MINIMUM. m 

44. Show that the right cylinder of given surface and maximum volume has its 
height equal to the diameter of its base. 

45. Show that the right cone of maximum entire surface inscribed in a sphere 
of radius a has for its altitude (23— Vi7)a/i6 ; while that of the corresponding 
right cylinder is (2 — 2/ V$)*a. 

46. Show that the altitude of the cone of least volume circumscribed about a 
sphere of radius a is 4^, and its volume is twice that of the sphere. 

47. The altitude of the right cylinder of greatest volume inscribed in a given 
sphere of radius a is la/ V3. 

48. The corner of a rectangle whose width is a is folded over to touch the 
other side. Show that the area of the triangle folded over is least when \a is 
folded over, and the length of the crease is least when \a is folded over. 

49. Show that the altitude of the least isosceles triangle circumscribed about an 
ellipse whose axes are la and 2d, is 3^. The base of the triangle being parallel to 
the major axis. 

50. Find the least length of the tangent to the ellipse x 2 /a 2 -\-y 2 /b 2 — I, inter- 
cepted between the axes. [Ans. a -j- ^.] 

51. A right prism on a regular hexagonal base is truncated by three planes 
through the alternate vertices of the upper base and intersecting at a common point 
on the axis of the prism prolonged. The volume remains unchanged. Show that 
the inclination of the planes to the axis is sec -1 V3 when the surface is least. 

[This is the celebrated bee-cell problem.] 

52. Show that the piece of square timber of greatest volume that can be cut 
from a sawmill log L feet long of diameters D and d at the ends has the volume 

2 LD* 



2J D - d 

53. A man in a boat offshore wishes to reach an inland station in the shortest 
time. He can row u miles per hour and walk v miles per hour. Show that he 
should land at a point on the straight shore at which 

cos a : cos fj = ti : v, 

approaching the shore at an angle a and leaving it at an angle /?. 

[This is the law of refraction.] 

54. From a point O outside a circle of radius r and centre C, and at a distance 
a from C, a secant is drawn cutting the circumference at R and R'. The line OC 
cuts the circle in A and B. 

Show that the inscribed quadrilateral ARR ' B is of maximum area when the 
projection of RR' on AB is equal to the radius of the circle. 

55. Design a sheet-steel cylindrical stand-pipe for a city water-supply which 
shall hold a given volume, using the least amount of metal. The uniform thickness 
of the metal to be a. 

If // is the height and R the radius of the base, then H = R. 

56. If a chord cuts off a maximum or minimum area from a simple closed curve 
when the chord passes through a fixed point, show that the point must bisect the 
chord. 



PART II. 

APPLICATIONS TO GEOMETRY. 
CHAPTER IX. 

TANGENT AND NORMAL. 



86. The application of the Differential Calculus to geometry is 
limited mainly to the discussion of properties at a point on the curve. 
Of chief interest are the contact problems, or the relations of a pro- 
posed curve to straight lines and other curves touching the proposed 
curve at a point. The application of the Calculus to curves is best 
treated after the development of the theory for functions of two 
variables. 

87. The Tangent (Rectangular Coordinates). — Lety =/{x), or 
cp{x,y) = o, be the equation to any curve. The equation to the 

secant through the points x, j/ and 
x\ , y x on the curve is 

y — y _ y x — y 









B, 


y 






^Px^ 




X,Y^ 








/o' 
















X - x -v, - x' 



(1 



X, F being the coordinates of an 
arbitrary point on the secant. By 
definition, the tangent to a curve 
at P is the straight line which is 
the limiting position of the secant 
PP 1 when P r ( = )P. But when 
= )y. The member on the right of 
equation (1), being the difference-quotient of y with respect to x, has 
for its limit the derivative of y with respect to x. At the same time 
the arbitrary point X, Y on the secant becomes an arbitrary point 
Therefore we have for the equation to the tangent 



Fig. 17. 
P x { — )P we have .r 1 ( = )jf and y x ( 



on the 
at P 



X - x 



= ¥-= Dy 



(»> 



in terms of the coordinates x, y of the point of contact. 



Art. 88.] TANGENT AND NORMAL. 113 

The equation to the tangent (2) can be written 

r- y={ x-, ) ^, (3 ) 

or in differentials 

(F-y) dx - (X - x)dy = o, (4) 

or in the symmetrical form 

X - x F-y 



dx dy 



(5) 



EXAMPLES. 

1. Find the equation to the tangent to the circle x 2 -\- y 2 = a*. 
Differentiating, we have 

2x -\- 2y Dy = o. 

.•. Dy — — x/y, and the tangent at x, y is 

Y-y + (X-x)j = Yy + Xx - (*■ + y 2 ) = o, 

or Yy -f Xx = a 2 . 

2. Find the tangent at x, y to x 2 /a 2 -f- y 2 /b 2 = 1. 

3. Find the tangent at x, y to x 2 /a 2 — y 2 /b 2 = I. 

4. Find the tangent at x, y to y 2 = \px. 

5. Find the tangent at x, y to x 2 -f y 2 -j- 2/y -(- 2£X -{- d = O. 

6. Show that the equation to the tangent at x, y to the conic 

(p{x, y)~ ax 2 -\- by 2 -\- 2/ixy -\- 2fx -\- 2gy -\- d = o 
is (ax + hy +/)X+ (hx + by + g)Y + ( /* + gy + d) = O. 

7. Show that the equation to the tangent at x, y to the curve 

x™-iX , y**-*Y 

is h — 7 ■ = I- 

^w b m 

8. Find the tangent at x, y to x 5 = a 3 j 2 . [5^7-^ — 2 Y/y = 3«] 

9. The tangent at x, y to x 3 — $axy -f- j 3 — o is 

(;/2 _ ^ x )K_j_ (^2 _ ^A^ _ ^. 

10. Find the equation to the tangent to the hypocycloid 

x* -{-y * = J, 

and prove that the portion of the tangent included between the axes is of constant 
length. 

88. If the equation to a curve is given by 

* = 0(*)i ^ = #(')« 

then, since dx = cp\t)dt, dy = ip / (f)df, we have for the equation 
to the tangent 

\F-y)<p'{t) = {X-x)ip\t). (1) 



114 APPLICATIONS TO GEOMETRY. [Ch. IX. 

EXAMPLES. 

1. If the coordinates of any point on a curve satisfy the cycloid 

x = 0(0 — sin 0), y = 0(1 — cos 6), 
show that the tangent at x, y makes an angle \B with Oy, and has for its equation 
V — y = (X - x) cot |0. 

2. In like manner, if 

;r = c sin 26(1 -+- cos 20), _y = <: cos 26(1 — cos 20), 
the tangent makes the angle with Ox, and its equation is 
Y - y = (X - x) tan 0. 

89. The angle at which two curves intersect is denned as the angle 
between their tangents at the point of intersection. 

\{y = <p(x) and j/ = ty(x) are two curves, and these equations be 
solved for x and y, we find the coordinates of the points of intersection. 
If the curves intersect at an angle &?, then since <fi'(x) and ip'(x) are 
the tangents of the angks which the tangents to the curves make with 
Ox, we have 

tan oa = *'-**, . (1) 

The two lines cut at right angles when (f)' x ip f x = — 1. 

Ex. Show that x' 2 -\- y 2 = Sax and y\2a — x) = x z cut at right angles and 

at 45 . 

90. The Normal (Rectangular Coordinates). — The normal at a 
point of a curve is the straight line perpendicular to the tangent at 
that point. 

If B t and 6 n are the angles which the tangent and normal at a point 
make with Ox respectively, then since one is always equal to the sum 
of \tt and the other, we have tan 6 t tan 6 n = — 1. Therefore 

_ dx _ 

tan 6 n — — — — D y x. 

dy 

Hence the equation to the normal at x, y to a curve is 

F-y+{X-x)D v x = o ) (1) 

or (F-y)D x y+X-x=-.o, (2) 

or in differentials 

(r-y)dy +(X~x)dx=o, (3) 

where D x y or DyX must be found from the equation to the curve. 

EXAMPLES. 

1. The equation to the normal at x, y to x 2 /a 2 -\- y 2 /b 2 = 1 is 

x ~ y 



Art. 91.] 



TANGENT AND NORMAL. 



"5 



2. The normal at x, y to y m =r ax n is 

ny Y -j- mxX = ny' 2 -f nix 1 . 

3. Show that the tangent and normal to the cissoid 

y\2a — x) = x 3 , at x = a, are, 
at (a, a), y = 2x — a, 2y -f- x = 3a; 

at (a, — a), y -\- 2x = a, 2y = x — 3a. 

4. In the Witch of Agnesi, y(\d 2 -f- x 2 ) = Sa 3 , the tangent and normal at 
x = 2a, are 

x -\- 2y — 4<7, y — 2x — 3a. 

5. Show that the maximum or minimum distance from a point to a curve is 
measured along the normal to the curve through the point. 

Let a, fi be a point in the plane of a curve <p(x, y) = o. 

If 8 is the distance from a, (5 to a point x, y on the curve, then 

When this is a maximum or minimum, 

d8 2 = — 2(a — x)dx — 2(/3 — y)dy = o, 
which is the equation (3), § 90, to the normal through a, ft. 

91. Subtangent and Subnormal (Rectangular Coordinates) 

The portion of the tangent, FT, included between the point of 
contact, P, and the or-axis, is called 
the tangent-length. The portion of 2/ 
the normal between the point of con- 
tact and the Jf-axis is called the nor- 
mal-length. The projections TM and 
ilAYofthe tangent-length and nor- 
mal-length on the jr-axis respectively 
are called the subtangent and subnor- 
mal corresponding to the point P. 

If /, n, S t , S n represent the tangent-length, normal-length, sub- 
tangent, and subnormal respectively, then we have directly from the 
figure 




••y 



/£■ <=-4 +(!)'/£■ 



y 



dy 
~dx' 



-•4 +(£)'• 



S f is measured from T to the right or left according as S t is -f- 
or — , and S„ is measured from M to the right or left according as 
S n is -f- or — . 

EXAMPLES. 

1. Show that the subnormal in the ellipse x 2 /a 2 -\- y 2 /b 2 = I is 

S n ~ - b 2 x/a 2 . 

2, Show that S t in y = a x is constant. 



n6 APPLICATIONS TO GEOMETRY. [Ch. IX. 

3. In j 2 = 2mx, show that S M = m is constant. 

t - ~-\ 

4. In the catenary y = \a\e a -\- e a J , n = y 2 /a. 

5. Show that <p(x, y) = o must be a straight line if S t /S n is constant. 

6. Show, in the cissoid X* = (2a — x)y 2 , that 

S t = (2ax — x*)/{$a — x). 

7. Show that the circle x 2 -f- y 2 = a 2 has n constant. 

92. Tangent, Normal, Subtangent, Subnormal (Polar Coor- 
dinates). — Let/(p, V) = o be the equation to any curve in polar coor- 
dinates, ip the angle which the tangent at any point makes with the 
radius vector, and <p the angle which the tangent makes with the initial 
line. From the figure we have 




PM± OP 1 = p 1 = p + Ap, 

y, rr . psin Ad 

tan MPP = ——+ -r- ai 

1 p-f Ap — p cos Ad 

sin Ad 
AS ~Ad~~ 



Ap Ad 1 — cos Ad' 

When Ad( = )o, we have, passing to limits, 

dd 
tan</> = p — , (1) 

/I — COS X 
■ = jQ sin x = o, when x(=)o. 

Also, since = d -f- ip, we have 

fjD p d 4- tanfl 
tan0 = i-ptan^ p ^ 
_ p- f- tan dD e p 
~ D o — p tan 0' ^ ' 



Art. 92. J TANGENT AND NORMAL. 117 

Observe that (2) is the same value as that obtained for D^y in 
§56. 




Fig. 20. 



Draw a straight line through the origin perpendicular to the 
radius vector, cutting the tangent in T and the normal in TV. We 
call /Wand PT, the portions of the normal and tangent intercepted 
between the point of contact, P, and the perpendicular through the 
origin, O, to the radius vector, OP, the polar normal-length and polar 
tangent-length respectively ; and their projections, OiVand OT, on this 
perpendicular are called respectively the polar subnormal and subtangent. 

We have directly from the figure 

/ = p sec = p 4/1 + p\D P e)\ (3) 

n = p esc ip =Vp 2 -\- (£>ep) 2 ' (4) 

S t = p tan ip = p 2 D p 6, S n — p cot ^ = Z> d p. (5) 

W T hen /?p^ is positive (negative), S t is to be measured from to 
the right (left) of an observer looking from to P. 

Putting p' =.D e p, we have for the perpendicular from the origin 
on the tangent 

(6) 









P = 


P 2 




iV + p' 2 


since 


pt = pS t . 


This can be written 








1 


= -+(£)' 


if we 


put p = 


i/u, 


for then 
dp 

dS 


I dfo 
= ~~7»d6' 

EXAMPLES. 



(7) 






1. In the spiral of Archimedes p = aQ, show that tan if; = Q, and S M is 
constant. 

2. Show that St is constant in the reciprocal or hyperbolic spiral p9 = a. 



nS APPLICATIONS TO GEOMETRY. [Ch. IX. 

3. In the equiangular spiral p = ae^ cot o-, show that ip = a, S t = p tan a> 
S n = P cot a. 

4. If p = a&, show that tan ip = (log a)— *. 

5. Show that the perpendicular from the focus to the tangent in the ellipse 

(I — e cos 0)p = a{\ — <? 2 ) 

I — e 1 

is £ 2 = pa 2 . 

r ' 2a — p 

6. Determine the points in the curve p = <z(i -)- cos 6), the cardioid, at which 
the tangent is parallel to the initial line. 

7. If p = a[l — cos 0), show that 

if; = £0, p = 2a sin 3 £0, ^ = 2a sin 2 10 tan £0. 

EXERCISES. 

1. Show that in .r, v) = o, the intercepts of the tangent at any point x, y on 
the axes are 

Xi — x- y D y x, Yi — y - xD x y. 

2. The length of the perpendicular from the origin on the tangent is 

x Dv - v 
P~ 



Vi + {Dyf 

3. Show that when the area of the triangle formed by the tangent to a given 
curve and the axes of coordinates is a maximum or a minimum, the point of con- 
tact is the middle point of thehypothenuse. 

Indicate D x y by y', and the area by £1. Then 

2n=x i Y i = - ^'- l x ' v X 
y 

Also, 

2 dD. = (y - xy')(y + xy')y" 
dx y' 2 

where y" = D* x y. For a maximum or a minimum D£l = o. The conditions 

y' ^ o, y" ^ o. y - xy' ^ o, y -f xy — o 
show, by Ex. i, that Xi = 2x, Yi = 2y. 

4. Find when the area of the triangle formed by the coordinate axes and the 
tangent to the ellipse 

is a minimum. 

5. Show that the tangent at the point (2, — i) of the curve 

x 3 + 2x\v - 3y°- + 4-r + }' ~ 4 = o 
is 8^-f- 15.1' = i. 

6. The line ex -\- y = e(i + it) is tangent to the curve 

sin x — cos .v = log y, at (it, e). 

7. The line;- + I = o is tangent at (+ I, — i) to 

x* _ 2.v-r 2 - 31' 3 + +n< + 4-v + 5)' + 3 = o. 

8. Determine the points at which the tangents to 

x 3 -f >' 3 = 3 x 
are parallel to the coordinate axes. (x = o, y = o), (.v = ± i, y = ± \*2). 



ART. 92.] TANGENT AND NORMAL. 119 

9. At what point of a a -f- 4,1' — 9 = o is the tangent parallel to x — y = o? 

(x = - \,y = 2.) 

10. The tangents from the origin to 

X i _ JJ,3 _[_ yPy _|_ 2x y1 _ Q 

are y — o, 3.V — y = o, jt -f- y = o. 

11. The perpendicular from the origin to the tangent at x, y of the curve 

jf3 _|_ yi — al is / =; \/axy. 

12. Show that the slope of the curve x 2 y 2 = « 3 (jf -(- /) to the x-axis is \tc 
at o, o. 

13. If x,y are rectangular coordinates and p, Q the polar coordinates of a point 
on a curve, show geometrically that when D x y — o we have D 9 p = p tan 0, 
and verify from the formulae in the text. 

14. Show that the curves 

x 2 y i X 2 y i 

*+P= s and ^+ v> =1 

cut at right angle if a 2 — b' 1 — a' 2 — b" 1 . 
iii 

15. In the parabola x 2 -J- j 2 = d 1 . show that at x, y the tangent is 

Xyi + Yx* = (axyf, 
and that the sum of its intercepts is constant and equal to a. 

16. The tangent at x, y to (x/a) 2 -f {yjbf = lis 

Xx/a 2 +(Y+ 2 y)/ Z b*y* = I. 
Also find the normal. 

17. The tangent and normal to the ellipse 

x 2 -j- 2y 2 — 2xy — x = o 
at x = 1 are, 

at (1, o), 2y = x — 1, _y -j- 2* = 2; 
at (1, 1), 2y = x -\-i, y -{- 2x = 3. 

18. In the curve y(x — i)(x — 2) = .*• — 3, show that the tangent is parallel 
to the x-axis at x = 3 ± |/2. 

19. In the curve (x/a) 3 -j- (y/bf = 1, show that (see Ex. I.) 

JQ ^ _ 
~a^ + I2" ~ X * 

20. Show that the tractrix 



c . c 



+ i / c 2_y2 = Jog 



\/c 2 -y 2 



+ V < 2 - y* 

has a constant tangent-length. 

21. In the curve y n = a n ~ l x, find the equation to the tangent; and determine 
the value of n when the area included between the tangent and axes is constant. 

22. In p{ae9 -j- be-«) = ab, show that 

S t = — ab/(aee _ te-8). 

23. If p 2 cos 20 = a 2 , show that sin if) = a 2 /p 2 . 

24. If two points be taken, one on the curve and one on the tangent, the points 
being equidistant from the point of contact, show that the normal to the curve is the 
limit of the straight line passing through the two points as they converge to the 
point of contact. 



120 APPLICATIONS TO GEOMETRY. [Ch. IX. 

25. If Q, f\ P are three points on a curve, /'the mid-point ot die arc QP. and 
Fthe middle point of the chord QP, show that the normal at P is the limit of the 
line /'Fas Q(=)P, P( = )P. 

26. Prove that the limit of any secant line through any two points P, Q on a 
curve is the tangent at a point .Pas P( = )P, Q( — )P. 

27. Show that as a variable normal converges to a fixed normal, their intersec- 
tion converges, in general, to a definite point, and find its coordinates. 

Let ( Y - y)y' -f X - x = o 

and ( Y - y x )y[ +JT-x 1 =o, 

where y', y[ represent D x y at x, y and x v y x , be the equations of a fixed normal at 
x, y and a variable normal at x xi y v Eliminating X, we have 

Y{/i - /) = y\y\ - yy' + - r i - *> 

= yi(y[ - y') + /to -y) + ( x i - *)• 



Y. 


= >'! + - 


~^~ x x — X 

~y'x -y' 




I 

= y + - 


y" 


X 


= x -y 





when x 1 ( = )^. 
Also, 

_ d 2 y 

where /' = -— -. 
y ax 1 

This point is the center of curvature of the curve for x, y. 



CHAPTER X. 
RECTILINEAR ASYMPTOTES. 

93. Definition. — An asymptote to a curve is the limiting position 
of the tangent as the point of contact moves off to an infinite dis- 
tance from the origin. 

Or, an asymptote is the limiting position of a secant which cuts 
the curve in two infinitely distant points on an infinitely extended 
branch of the curve. 

94. We have the following methods of determining the asymptotes 
to a curve f{x, y) = o: 

I. The equations to the tangent at x, y and its axial intercepts 
are 



Y- 


-y = 


,{X 


'dx' 




Fi = 


^y - 


A, 

dx 






-. X — 


dx 



If we determine, for x =y = 00 , 

then the equation to the asymptote is 

x y 

Or, if we determine, for x =y ■— 00 , 

tjx-- m > 

and either a or b as above, we have for the asymptote 

y 

y = mx 4- b or x — \- a. 

m 

This method involves the evaluation of indeterminate forms, 
which must be evaluated either by purely algebraic principles or by 
aid of the method of the Calculus prescribed for such forms. The 
algebraic evaluations are of more or less difficulty, and another 
method will be given in III for algebraic curves. 

121 



J22 APPLICATIONS TO GEOMETRY. [Ch. X. 



EXAMPLES. 

x 2 y 2 
1. Find the asymptotes to the hyperbola — — — = i. 

a 1 b 2 

We have A" t - = a 2 /x, Yi — — b' l /y. These are o when x — y = oo 
Therefore the asymptotes pass through the origin. Also, 

dy b 2 x b i 

= 3== ± 



<rfr a 2 y a ^/ l _ a -i/ x -i ' 

the limits of which are ± b/a when .r =r oc . The equations to the two asymptotes 
are ay = ± bx. 

2. Find the asymptotes to the curves 

(a), y =^ log x. x = o. 

(b). y = ex. (Fig. 33.) y = o. 

(c). y = e~* 2 . (Fig. 34.) y = o. 

(d). y 2 e 2 * = x 2 — 1. y = o. 

(<?). 1 + j = ,?*. .* = o, y — O. 

(/). y = tan ax t y = cot a.r, y = sec A*. 

3. Show that y = x is an asymptote of x 3 = (x 2 -f- 3tf 2 }y. 

4. x -|-j' = 2 is an asymptote of jy 3 = 6x 2 — x 3 . 

5. x = 2a is an asymptote of x 3 = (2a — x)y 2 . 

6. x 3 -|- y z = a z has / -J- x = o for an asymptote. 
7t The asymptotes of (x — 2d)y 2 = X s — a z are 

x =: 2a and a -|- x = ± > / * 
x = 2a is readily seen to be an asymptote. For the others express Dy in terms 
of x and make x = 00 ; the result is ± 1. Find the intercept in same way. 

8. Find the asymptote of the Folium of Descartes 

x* -\-y* = $axy. 

See Fig. 49. The asymptote is x -f- y -f- a =0. Putj = wx in the equation 
to determine slope and intercept. 

II. We can sometimes find the asymptotes to curves by expansion 
in a series of powers. Thus, if 

j-=v+«.+j+3+---» 

then y = a x -f a x is an asymptote. For, evaluating as in I, we 
have m = a Q , V i = a v 
Observe also, if we have 

> = *(*) +J+J+..., 

then when x = oc the difference between the ordinate to this curve 
and that of the curve y = (p(x) continually decreases as x increases. 
We say the two curves are asymptotic to each other. 



Art. 94-] RECTILINEAR ASYMPTOTES. 123 

EXAMPLES. 

9. In Ex. I, I, we have 

b I o 2 \¥ b I \ a 2 \ 

As x increases indefinitely, the point x, y converges to the straight-line asymptote 
ay — ± bx. 

10. Solve Ex. 3, I, by expansion. 

11. Solve Ex. 6, I, by expansion. Here we find that the given curve and the 
hyperbola 

y 2 = x 2 -|- 2ax -\- 4a 2 
have the same asymptotes. 

III. We pass now to the most convenient method of determining 
the asymptotes to algebraic curves. 

If the given curve is a polynomial, /"(.v,;) = o, in x and;', or 
can be reduced to that form, we can always find its asymptotes as 
follows: 

Rule 1. Equate to o the coefficients of the two highest powers 
of x in 

/{x, mx -f- b) = o. 

These two equations solved for m and b furnish the asymptotes 
oblique to the axes. 

Rule 2. Equate to o the coefficients of the highest powers of x 
and of y \vif{x, y) = o. The first furnishes all the asymptotes parallel 
to the jf-axis, the second those parallel to the_y-axis. 
Proof: (A). The straight line 

y — mx -\- b ( 1 ) 

cuts the curve 

f(x,y) = o (2) 

in points whose abscissas are the values of x obtained from the solu- 
tion of the equation in x, 

f(x, mx -f b) = o. (3) 

If (2) is of the nth degree in x and y, then (3) is of the nth 
degree in x, and will furnish, in general, n values of x (real or 
imaginary). 

Let (3), when arranged according to powers of x, be 

A n x" + A n _ x x^ + . . . + A x x + A = o. (3) 

If one of the points of section of (1) and (2) moves off to an 
infinite distance from the origin, then one root of (3) is infinite, and 
the coefficient, A M , of the highest power of x must be o, or A„ = o. 

This is readily seen to be true by substituting i/z for x in (3), 
and arranging according to powers of z. Then when z = o, we have 
x = 00 , and A n = o. 

In like manner if a second point of intersection of (1) and (2) 
moves off to an infinite distance on the curve, a second root of (3) 



124 APPLICATIONS TO GEOMETRY. [Ch. X. 

is infinite and we must have the coefficient of .v"^ 1 equal to o, or 

A» , = O. 

When (i) and (2) intersect in two infinitely distant points, then 
(1) is an asymptote of (2), and we have for determining the 
asymptotes the two equations 

A„ = °> A n-t = O. 

These two equations when solved for m and b give the slopes and 
intercepts on thej>-axis of the oblique asymptotes of (2). 

EXAMPLES. 

12. Consider x 3 = (x 2 -f- 2> al )}'< see Ex. 3. 
Here x 3 — x 2 y — 3a 2 y — o 

becomes (1 — m)x 3 — bx l — $a 2 mx — ^a 2 b = o, 

when mx -\- b is substituted for y. Hence 

I — m = O and — b = o 
give y = x as the oblique asymptote. 

13. In x 3 -)- y z = 2>°xy, Ex. 8, put y = mx -j- £. 

.-. (1 -\- m 3 )*? -f yn{mb — a)jr 2 -j- . . . = o, 
it being unnecessary to write the other terms. 

Hence m = — I, /; = — (7. Therefore the oblique asymptote is 
y ■=. — x — a. 

14. Show that j = x -(- £# is an asymptote of y z = ax 2 -f- x 3 . 

15. The asymptotes of y* — x 4 -f- 2ax 2 y = b 2 x 

are y ■=. x — \a and y -j- x -\- \a = o. 

16. x 3 + 3x 2 j - x)' 2 - 3/ + x 2 - 2xy + 3/ 2 -f 4r -f 5 = o 
has for asymptotes 

y + i* + f = o. 7 = x -f- i, ;+-«• = !• 
(B). If the term ^4 M _ 1 ^*' -1 is missing in (3), or if the value of m 
obtained from A n = o makes A n _ x vanish, then (3) has three infinite 
roots when 

A n = o and A n _ 2 = o, 
which equations give the values of m and b which furnish the 
asymptotes. A n _ 2 will be of the second degree in b, furnishing two 
b's for each m, and there will be for each m two parallel oblique 
asymptotes, which we say meet the curve in three points at 00 . 

If also the term A n _ ? x n ~ 2 is missing, or if A n _ 2 vanishes for the 
value of /;/ obtained from A n = o, then the equations 

A n = o, A n _ 3 = o 
furnish three parallel oblique asymptotes, in general, for each m. 

EXAMPLES. 

17. If (x + r) 2 (.r 2 + y 2 -f xy) = a 2 y 2 -f a*(x - y), 
then A n _ x — (1 -j- vif{\ -\- m -f m 2 ), 

A H - t - o, 
An_ 2 = b 2 - a 2 . 
.*. m = — 1, b = ± a give asymptotes y = — x ± a. 



Art. 95.] RECTILINEAR ASYMPTOTES. 125 

18. In x*{y + xf + 2ay\x + y) + $a 2 xy + a*y = o, 
the asymptotes are y -|- x = 2^7, _y -)- x -(- 4^ = o. 

19. Find the asymptotes to the curves: 

(a), xy' 2 — x*y = a 2 (x -j- y) 4 P. x — o, y — o, x — y. 
(b). y* — X s = a 2 x. y — jr. 

(r). x* — j' 4 = <7-\rj' -f- b 2 y 2 . x -\- y = O, .r = ;'. 

(C). For the asymptotes parallel to the coordinate axes, the fol- 
lowing simple process determines them : 

Arrange f(x, y) = o according to powers ofy, thus: 

Ay -f (^ + c)y-> + (^ + Gx + #)y- 2 + . . . = o. ( 4 ) 

If the highest power ofy is, ?i, the degree of the curve, there will 
be no asymptote parallel to Oy, since then A ^ o. If, however, the 
term Ay n is missing, or A =0, then for any assigned x one root in 
the equation (4) in y will be 00 . If, now, Bx -f- C — o, a second 
root of (4), in >', is 00 at a: = — C/B, and this will be an asymptote 
to the curve, since D x y is 00 for the same value of x which makes 
y = 00 . 

If the terms involving the two highest powers of y in (4) are 
missing, then 

Fx 2 + Gx + H = o 

makes three roots of (4), in j', infinite, and this is the equation to two 
asymptotes parallel to Oy, and so on generally. 

In like manner, arranging f{x, y) according to powers of x, we 
find the asymptotes parallel to Ox by equating to o the coefficient of 
the highest power of x. 

Therefore the coefficients of the highest powers of x and y in the 
equation to the curve, equated to zero, give all the asymptotes parallel 
to the axes. Of course, if these coefficients do not involve x or y 
they cannot be o and there are no asymptotes parallel to the axes. 

EXAMPLES. 

20. Find the asymptotes to the following curves: 

(a). y 2 x — ay 2 = X s -\- ax 2 -j- P. x = a, y = x -f- a, y -\- x -\- a = o. 

(b). y(x 2 — %bx -\- 2d 2 ) = x 3 — ^ax 2 -\- a*, x = 6, x = 26, y -f- Z a = x -f 3^- 
(c). x 2 y 2 z= a\x 2 -j- y 2 ). x = ± a, y = ± a. 

(d). x 2 y 2 - a\x 2 - y 2 ). y -f a = o, y — a = o. 

(e) y 2 a = y 2 x -\- x 3 . x = a. 

(/). (x 2 -y 2 ) 2 - 4y 2 +y = o. 

(g). x 2 (x - yf - a\x 2 +y 2 ) = o. 

(h). x 2 (x 2 + a 2 ) = (a 2 - x 2 )y 2 . 

if). x 2 y 2 - jt 5 4. x 4- y. 

(/). x 2 y 2 =\a+y)\b 2 -y 2 ). 

(£). y{x - yf - y( X - y) + 2. 

95. Asymptotes to Polar Curves. — If f(p, 6) = o is the equa- 
tion to a curve in polar coordinates, then, when it has an asymp- 



126 APPLICATIONS TO GEOMETRY. [Ch. X. 

tote, that asymptote must be parallel to the radius vector to the point 
at co on the curve, if the asymptote passes within a finite distance of 
the origin. 

The distance of the asymptote from the origin is the limiting value 
of the polar subtangent when the point of contact is infinitely distant. 

To determine the polar asymptotes to /{p, (J) = o, determine 
the values of which make p = co . These values of give the 
directions of the asymptotes. 

If the equation can be written as a polynomial in p, the values of 
are furnished by equating to o the coefficient of the highest power 
of p. 

To construct the asymptote when, — a, the direction has been 
found; evaluate for 0( = )a and p = co the subtangent 

»(=)o 

where pu = i. The perpendicular on the asymptote is to be laid off 
from the origin to the right or left of an observer at the origin look- 
ing toward the point of contact, according as / is -\- or — respec- 
tively. 

EXAMPLES. 

21. Let p = a sec + b tan 0. 

p = oo when = \Tt; also, 

2 dO __ (a + b sin 0) 2 

dp ~ a sin -J- b ' 

the limit of which is a -f b. The asymptote is then perpendicular to the initial 
line at a distance a -f- b to the right of O. Also, when = §tt, p = oo , and the 
corresponding value of the subtangent gives a — b and another asymptote. 

22. Show that p 2 sin (0 — a) -\- op sin (0 — 2a) -\- a 2 — o has the asymp- 
totes p sin (0 — a) = + a sin a. 

23. Find the asymptotes of p sin = aO. 

24. Find the straight asymptotes of p sin 4.B = a sin 3©. 

25. Show that p cos = — a is an asymptote of p cos = a cos 20. 

26. b = (p — a) sin has p sin = /; for asymptote. 

27. Determine the asymptotes of p cos 20 = a. 

Polar curves may have asymptotic circles or asymptotic points. 

EXAMPLES. 

28. Find the asymptotes of pO = a, for = o, = oo . Fig. 57. 

29. Find the circular asymptotes of p{Q -j- a) = bQ. and of 

rt-0 2 _ gQ 2 _ + cos 

P - efil 2 ' P ~ 4-sin0' P " + sin 0' 



CHAPTER XI. 
CONCAVITY, CONVEXITY, AND INFLEXION. 

96. On the Contact of a Curve and a Straight Line. — Let 

y =f(x) be the equation to a curve. The equation to the tangent, 
§87, at x = a (Y being the ordinate corresponding to x) is 

r =/(<*) + (* - «)/"(«). 




The difference between the ordinates of the curve and tangent at 
any point (by the theorem of mean value) is 

/(x)-r=^x-a)Y"{S). 

If f"{a) 7^ o, this difference will retain its sign unchanged for 
all values of x in the neighborhood of a. Therefore throughout this 
neighborhood the curve will lie wholly on one side of the tangent. 
It will lie below the tangent when f"{a) is — , and above it when 
/» is +. 

The curve y — f{x) is said to be co?icave at a when /"(a) is 
negative, or the curve lies below the tangent there; and is said to be 
convex at a when f"{a) is positive, or the curve lies above the tan- 
gent there. 

EXAMPLES. 

1. The curve y ■=. e x is always convex, since D 2 e x = e x is always positive. 

2. The curve y = log x is always concave, since D 2 log y = —x~ 2 is always 
negative. 

3. The curve y = x 3 -\- ax is convex when x is positive and concave when x 
is negative, since D 2 y = 6x. 

127 



128 



APPLICATIONS TO GEOMETRY. 
Points of Inflexion. 



[Ch. XL 





y 






\ 


p 


^M ' 


A/ 


' 


a 


; a 


a' 



Fig. 22. 

Suppose, at x = a, we have f"{a) = o, but /'"(a) ^ o. Then 
the difference between the ordinates of the curve and tangent at a is 

Ax)-r=\(x-a)Y'"(S). 

Since /'"{a) 9^ o, then throughout the neighborhood of a, f"\B,) 



keeps the same sign as its limit f'"{d). But (x 



) 3 changes from 

- to -f- as x increases through a. Consequently the corresponding 
point P on the curve crosses over from one side of the tangent to the 
other as P passes through A. 

The curve is convex on one side of A and concave on the other. 
The curve is said to have a point of inflexion at x,y when at this 
point we have D x y determinate and D l y = o, Dy ^ o. 

At a point of inflexion x = a a curve is said to be convexo- 
concave when it changes from convex to concave as x increases 
through a, and to be concavo-convex when it changes from concave 
to convex as x increases through a. See the points A and A' in 
Fig. 22. 

EXAMPLES. 

4. If y — 2{x — af -f- 4_r — i, 

y" ■==. \z{x — a) = o, when x — a, 

and y'" = 12. 

The curve has a concavo-convex inflexion at x = a. 

5. Show that every cubic 

f(x) = ax* -f dx 2 -f ex + d 
has an inflexion and classify it. 

Again, suppose at x = a we have 

/"(a) = o, /'"(a) = o, 
Then 

(x - a) 



f"(a) 9* o. 



A*) 



4! 



/ ,v (£)- 



In the neighborhood of a, f lv {£i) keeps its sign unchanged, as also 
does (x — ay. Consequently the curve lies wholly on one side of 
the tangent, and is convex or concave according a.sf iv (a) is -J- or — . 

In general, if /"(a) = . . . . =/'"(") = °»/ m+1 ( a ) ^ °> then 



Art. 96.] CONCAVITY, CONVEXITY, AND INFLEXION. 129 

If /;/ -f~ 1 is even, the curve is concave or convex at a according as 
f"" l {a) is negative ox positive. 

If ?n 4- 1 is odd, the curve has an inflexion at x = a, and is concavo- 
convex if/"" l+1 (a) is -|-, and is convexo-concave \i'f" t+l (a) is — . 

The tangent at a point of inflexion is sometimes called a station- 
ary tangent, since D x 6 = o there. For, 6 being the angle which the 
tangent makes with Ox, we have tan 6 — D x y, etc. 

The conditions for a point of inflexion given above, fory^v, y) = o, 
are exactly those which have been previously given for a maximum or a 
minimum of D^y. For y = f(x) has a convexo-concave inflexion 
whenever f(x) is a maximum, and a concavo-convex inflexion whenever 
f'{x) is a minimum. The investigation of y —f(pc) for points of 
inflexion amounts to the same thing as investigating the maximum 
and minimum values oiy =■ f'{x). 

It is not necessary to give many examples of finding points of 
inflexion, since it would be but repeating the work of finding the 
maximum and minimum values of functions. 

EXAMPLES. 

6. Show that x 3 = (a* -j- x 2 )y has an inflexion at the origin. What kind of 
inflexion ? 

7. Show that a 3 y = bxy -j- ex 3 -f- </x* inflects at o. o. 

8. The origin is an inflexion on a m ~ 1 y = x ni , if m > 2 is an odd integer. 

9. When is the origin an inflexion on y n = kx m ? 

10. Find the point of inflexion on x* — T>bx 2 -j- a 2 y ~ o, and classify it. 

[x — b,y = 2& 3 /a?.] 

11. Show that the inflexions on/(p, 6) = o are to be determined from 



td P y d^p 



See § 56. If we put p = i/u, this takes the simpler form 
u -\- u'q = o. 

The polar curve is concave or convex with respect to the pole according as 
u 4- u'q is -|- or — . The curve in the neighborhood of the point of contact is con- 
cave or convex with respect to the pole according as it does or does not lie on the 
same side of the tangent as the pole. 

12. Find the inflexion on p sin 8 = aQ. 

13. In pB nt = a there is an inflexion when = \/m(i — m). 

14. Find the points of inflexion on the curves: 

(a), tan ax — y. (d). y = e~ a x • 

(b). y = sin ax. (e). y = (x — i)(x — 2)(x — 3}. 

(c). y = cot ax. (/). p(B 2 - 1) = aQ 2 . 

15. Show that the curve x(x 2 — ay) = a 3 has an inflexion where it cuts Ox. 
Find the equation to the tangent there. 

16. Show that x 3 -\- y 3 = a 3 has inflexions on O x and O y . 

17. The inflexions of x 2 y = a 2 (x — y) are at x = o. x ■=. ± a \/$. 

18. x = log (y/x) inflects at x = — 2, y = — 2e~ 2 . 

19. PG 2 = a has an inflexion at p — a 4/2. 



CHAPTER XII. 
CONTACT AND CURVATURE. 

97. In the preceding chapter we have studied the character of the 
contact of a curve with its straight-line tangent. Now we propose 
to study the nature of the contact of two curves which have a 
common tangent at a point. 

98. Contact of Two Curves. 

I. Let^' = <p{x) and y = xp(x) be two curves, the functions 
and tp having determinate derivatives at a. 

If we solve y = <p(x) and y = t/>(x) for x and y, we find the 
points of intersection of the curves. 

(1). If (p(a) = ip{a) and <p'(a) 7^ ip\a), the curves cut at a, and 
cross there. For, by the law of the mean applied to the function 

F(x) = 4>{x) - f( X ), 
we have 

<t>(x) - i'(x) = (x - a)[<P'(S) - f(S)l (§ 62) 

The derivatives 0'(£), ip'(&) are arbitrarily nearly equal to <fi'{ci) 
and ip'(a) for x in the neighborhood of a. Therefore, since 
<p\a) t^ $'{a), the difference / (<?) — tp'(£) keeps its sign unchanged 
in the neighborhood of a, and x — a changes sign as x passes 
through a. 

(2). If we have <p(a) = f(a), <p'(a) = ip'(a), but (p"{a) j* ip f \a), 
then the curves have a common tangent at a, and are said to be tan- 
gent to each other, and to have a contact of the first order at a. 

By the law of the mean, the difference 

VH-v) - f(x) = i(x - a?[4>"(£) - *"(«)] 
shows that this difference does not change sign as x increases through 
#, and therefore the curves do not cross at a. 

(3). If 0(a) = i/:(a), 0'(,z) = J,' (a), 0" '(a) = 0"(«), but 0'» 
t* ip"'{a), then the curves have a contact of the second order at a] 
and we have 

<P(x) - f(x) = i,(x - a)H<p"'(£) - <&'"(£)]. 

This shows that the curves do cross at a, since the difference of their 
ordinates changes sign as x increases through a. 

(4). In general, if 0(~r) and tp(x) and their first n derivatives at a 
are equal, but their (n -j- x ) tn derivatives are unequal, then the 

130 



Art. 07- ] CONTACT AND CURVATURE. 131 

curves are said to have an «th contact at a, or a contact of the n\\\ 
order. They do or do not cross at the point of contact according as 
n -J- 1 is odd or even. 

For we have, by the law of the mean, 

0(.v) - f{x) = ( ^*yV +1 (g) - r +, (s)i 

which changes sign or does not according as n -f- 1 is odd or even 
when x increases through a. 

Two functions are said to have a contact of order n at a value of 
the variable when for that value of the variable the corresponding 
values of the functions and their first n derivatives are equal. 

II. The character of the contact of two curves is made clear by 
the following theorem: 

If two curves y = <p(x) and y = zp(x) intersect in n distinct 
points a£ a x , a 2 , . . . , a„, then when these n points of intersection 
converge to one point, the curves have a contact of order n — 1. 

To prove this the following lemma* will be established: 

If F(x) vanishes at a x , a 2 , . . . , a M , then 

F{x) = (■*-",)--(*-*.y (g)) 

where B, is some number between the greatest and least of the num- 
bers x, a x , . . . , a n . 

Consider the function of z, 

J{z) = (x - a x ) . . . (x - a n )F(z) - (z - a x ) . . . (0 - a n )F(x). 

We havey(z) = o at the n -f- 1 values of 2 equal to x, a 1 , . . . , # M . 
By Rolle's theorem, J\z) vanishes n times, once between each con- 
secutive pair of these numbers. Also by the same theorem J"{z) 
vanishes n — 1 times, once between each consecutive pair of numbers 
at which J'{z) vanishes; and so on, until finally /"(z) vanishes once 
between the pair of values for which J n ~ x {z) vanishes. This value, 
say £, at which J n {z) vanishes is certainly between the greatest and 
least of x, a , . . . , a n . Hence 

/•(£) = (x -a x )... (x - a H )F»($) - n\ F(x) = o, 
and the lemma is proved. 

Now let F(x) = <p(x) — ip(x). Then 

cP{x) - Hx) = ^- a ^' n ; {X " a '\ ^^ - **(*)]. 

This shows that when a x = a 2 = . . . = a n = a, we have 

<mx) - fix) = ( *~ aV V (g) - rm, 

where B, lies between x and a. 

* Due to Ossian Bonnet. 



132 APPLICATIONS TO GEOMETRY. [Ch. XII. 

This last equation shows that 0(.f) and ip(x) and their first ?i — 1 
derivatives at a are equal, or the two curves have a contact at a of 
order n — 1. Therefore, when two curves have a contact of the «th 
order, it means that they have ?i -\- 1 coincident points in common 
at a; or, as we sometimes say, they intersect in n -\- 1 consecutive 
points. A curve which cuts another n times in the neighborhood of 
a point, leaves that curve on the same side it approaches it when n 
is even and leaves on the opposite side when n is odd. Thus we see 
why it is that curves having even contact cross, while those having 
odd contact do not cross, at the point of contact. 

99. To find the order of contact of two given curves, we must 
solve their equations for the points of intersection, and compare 
their corresponding ordinate derivatives at these points. 

EXAMPLES. 

1. Find the order of contact of the curves 

y =. oP and y = 3X 2 — t> x + !• 
Solving the equations, we find that x = 1, y = I is a point common to both 
curves. Also, their first derivatives, Dy. are equal to 3 there, and their second 
derivatives, D 2 y, are equal to 6; while their third derivatives, D A y, are not equal 
to each other. Therefore, at the point 1, 1 the curves have a contact of the second 
order. 

2. Show that the straight line y = x — 1 and the parabola \y = x 2 have a 
first-order contact. 

3. Find the order of contact of 

gy = x 3 — 3-r 2 -|- 27 and gy -j- t,x = 28. [Second.] 

4. Find the orders of contact of the curves: 

(a), y = log (x — 1) and x 2 — 6x -f- 2y -f- 8 = o. [Second.] 

(&). ty = x 1 - 4 and x 2 + y 2 - zy = 3. [Third.] 

(c). xy = a 2 and (x — 2a) 2 -\- (y —2a) 2 = 2xy. [Third.] 

5. Find the value of a in order that the hyperbola xy = $x — I and parabola 
y = x -\- 1 -)- a{x — i) 2 may have contact of the second order. 

100. Osculation. — (i). We can always find a straight line which 
has a contact of the first order with a given curve y = <p(x) at a given 
arbitrary point. In general, at any point of ordinary position, a 
straight line cannot have a contact with a curve of order higher than 
the first. 

For, let y = mx -f- ^ be the equation to a straight line, in which 
m and b are arbitrary and are to be so determined that the straight 
line shall have the closest possible contact with the curve y = <fi(x) 
at x — a. 

Then we must have 

ma -\- b = (fi{a), 
m r= 0(tf)« 
These two conditions completely determine m and b, and give 
y= <f>(a) + (x - a)<p'(a) 



Ari. ioo.] CONTACT AND CURVATURE. 133 

as the equation to the required line, which has contact of the first 
order with the curve at a. This is the familiar equation to the tan- 
gent to the curve at a. 

The line can have no higher contact with the curve at a unless 
we have (p"(a) = o, and so on, see §98. At an ordinary point of 
inflexion the tangent has a contact of the second order, and cuts the 
curve there in three coincident points crossing the curve. 

(2). Consider the equation to the circle 

(x-ay + (r-/3)* = x\ (1) 

This is the most general form of the equation to the circle, and 
can be made to represent any circle whatever, by assigning proper 
values to the arbitrary constants a, /j, R, the coordinates of the 
centre and the radius. 

Let us determine <*, (3, and R, so that the circle shall have the 
closest possible contact with a given curves = cp(x) at a given point 
x, y of general position on the curve. 

Differentiating (1) twice with respect to X, 

X- a +(F-fi)DF=o, (2) 

i + (r-j})&r+(£ry = o. (3) 

The conditions for the contact are 

F=y, DF = cf>\x), £*F= 0"(*)'. 

The values of a, f3, R determined from the three equations 

(.r - af + (y- fif = &, (4) 

x — a + O - fS)(f>'(x) = o, (5) 

1 + O - A)0"(*) + [>"(*)] = o, (6) 

determine the circle of closest contact, of the second order, at x, y 

on the curve. Solving these equations and writing y f , y" for r , (p" ', 

we have for the coordinates of the centre of curvature 

1 -I- 1/ 2 1 4- v' 2 

P=y + ^-, a = x--y^L, (7) 

and for the radius of curvature 

(8) 



(1 +y*)i 



y 

Whenever the coordinates x, y are given, we can substitute in 
these formulae and compute a, f3, and R, and write out the equation 
to the circle. 

Observe that the three equations completely determine the circle, 
and the circle at a point of ordinary position on the curve can have 
no closer contact with the curve than that of second order. Observe 
that this is the same circle obtained in § 79, 111. (3), where we con- 
sidered the circle which was the limiting position of a circle through 
three points on the curve when these three points converge to x, y 



134 APPLICATIONS TO GEOMETRY. [Ch. XII. 

as a limit. Having a contact of the second order with the curve, 
the circle of curvature crosses over the curve at the point of contact. 

This circle is called the circle of curvature of the curve at the 
point x, y, and R is called the radius of curvature, the point a, /J 
is called the centre of curvature of the curve at x,y. 

(3). In general, when the equation of a curve y = tp[x) con- 
tains a number, n -f-i, of arbitrary constants, we can determine the 
values of these constants so that the curve shall have a contact of the 
?zth order with a given curve y = <p(x), at a given point of arbitrary 
position and no higher contact. For, if we equate the values of the 
function ip and its first n derivatives to the corresponding values of 
<p and its first n derivatives, we shall have n -\- \ equations between 
the n -f- 1 arbitrary constants in if). These equations serve to deter- 
mine the values of these constants which will make y = i/>(x) have 
an ?ith contact with y = <p(x) at the point under consideration. 
This is the highest contact such a curves = tp can have with a given 
curve y = (p at a point of ordinary position. Thenjy =: ip is said to 
osculate the curve jy = at x,y. 

At certain singular points an osculating curve can have a contact 
of higher order with a given curve than that which it has at a point 
of ordinary position — as, for example, the tangent line to a curve at 
an inflexion. 

101. Construction of the Circle of Curvature.— Since By is the 
same, at the point of contact, for the circle and the curve, they have 
a common tangent and normal there; also, the centre of curvature 
is on the normal to the curve. They have the same convexity or 
concavity at the point of contact. The radius of curvature, involv- 
ing the radical sign, is ambiguous; we remove the ambiguity by 
taking R as positive when_j/' is positive, or when the curve is convex ; 
and negative when y" is negative or the curve is concave. Conse- 
quently the value of R is 

y 

The center of curvature is to be constructed by measuring off R 
from the point of contact along the normal, upward or downward 
according as R or y" is -\- or — . 

EXAMPLES. 

1. Find the radius of curvature at any point on the parabola .r 2 = \my. 
Here 2my' = x, 2my" = 1, 1 -\- y' 2 =. 1 -\-y/m\ 

2 (m -f ; ■)•• 

••• P = H 7= — • 

ym 

2. Find the radius of curvature in the catenary 

y = *« (** + '"). 



Art. ioi.] CONTACT AND CURVATURE. 135 

Here /=\\e a — e « J , y" — y/d i ; .-. p = +y*/a. 

Show that the radius of curvature is equal and opposite to the normal-length. 
3. In the cubical parabola $a 2 y = a- 3 we have 

(1 +/*)* 

_ (a* -f x 4 )= 



2a 4 x 

4. Newton's Rule for the Radius of Curvature. At any point P on a given 
curve draw a circle tangent to the curve and cutting it in a third point Q at dis- 
tances/ and q from the common normal and tangent respectively. 

Let r be the radius of the circle. Then, by elementary geometry, the products 
of the segments of the secants are equal, and we have 

f = q(2r - q), 

2q ' 2 

When Q(=)P, the circle becomes the circle of curvature at /'and^r — P. 

... K=S*L. 

^ 2q- 
when /(=)o, q( = )o. 

5. If Q, P, R are three points on any curve, such that V is the middle point 
of the chord QP, and P is the mid-point of the arc QP, show that 

when Q(=)P, P{ = )P. 

EXERCISES. 

1. Find the parabola y = Ax' 2 -f- Px -j- C which has the same curvature as a 
given curve j = f{x) at a given point x. y. 

Y = fix) + (X - *)/'(*) + l(X - x)Y"(x). 

2. Show that a straight line has contact of second order with a curve at a point 
of ordinary inflexion. 

3. Show that the radius of curvature is 00 at a point of inflexion, and explain 
geometrically. 

4. Show that the circle of curvature has a contact of third order at a maximum 
or a minimum value of P; and therefore does not cross the curve at such a point. 

At a max. or min. value of P we have D x R l — o. Differentiating (8), § 100, and 
solving, we find for the curve 

Computing D%y, for the circle, from (5) and (6), we find it has the same value. 

5. Show from (5), § 100, that the normal passes through the center of curva- 
ture. 

6. Find the radius of curvature for the ellipse 

R = —ab ~ U + ¥) "^ 

<p being the eccentric angle. 



136 APPLICATIONS TO GEOMETRY. [Ch. XII. 

7. x 3 + y = a* is satisfied by x = a cos :{ 0, y — a sin 3 0. Show that 

R = — ziaxyf. 
y 

8. Show that the radius of curvature of e x = sec (x/a) is 

R = a sec (x/a). 

9. The coordinates of a point on a curve are 

x = c sin 2/(1 -j- cos it), y -— c cos 2/(1 — cos 2/); 
show that R = 4<r cos 3/. 

10. Find A' for X s = ay' 1 . 

11. Show that, when y = sin x is a maximum, i? = I. 

12. Find the center and radius of curvature of xy = a 2 . 

a = (3* 2 + f)/2x, (5 = (x* + 3.r 2 )/2>', ^ = (* 2 + y*f/ 2 a\ 

13. Show that if a variable normal converges to a fixed normal as a limit, their 
intersection converges to the center of curvature as a limit. 

The equations to the normals at x v y x and x, y are 

<"-*>£ + •*-* = * <-y-y)% + x-* = o. 

The ordinate of their intersection is 

d )\ dy , 

d _y_ _ dyy 

dx dx^ 

which takes the illusory form 0/0 for x x = x. 

When evaluated in the usual way, we have, when jr 1 (=)x, 

which is the ordinate of the center of curvature. 

Substitution of Y — y in the equation of the normal gives X as the abscissa of 
the center of curvature. 

14. Find the radius of curvature at the origin for 

2x 2 -j- $xy — \y 2 -f- x 3 — 6y = o. 
Using Newton's method, 

' Aj iy 2 

15. Find the radius of curvature at the maximum ordinate of y = e—^x." 1 . 
What is the order of contact of the circle of curvature ? 

16. If f(p, 6) = o is the polar equation to any curve, show that at any point 
p, G the radius of curvature is 

(f- + 2p" - pp'" 

where for brevity we write p' EE D$p, p" = D^p. 

This follows immediately from substituting for Dy and D l \\ (1) and (2\ £ 56, 
in (8), § 100. 

17. Show that if pn — I, u' = D e u, 11" = D%h, the value of the radius in 
Ex 16 becomes 

R = ( " 2 + U " 1)l 



\u + u") 



Art. ioi.] CONTACT AND CURVATURE. [37 

18. Since at a point of inflexion r" = o. we have there R = 00 . Therefore the 
inflexion condition for a polar curve is, as found before, u -(- it" = o. 

19. If p — nO, show that R = «(i + 6 2 ) i /(2 + 2 ). 

20. If p = a , then R = p[i + (log a) 2 ] 1 . 

21. If /o = 29 — II cos 2O, ^ = 00 at cos 20 = T » T . 

22. Show that R = lab, for p = a sin bQ, at the origin. 

23. Find the radius of curvature for the hyperbola 

x 2 /a 2 — y 2 /b 2 = 1. 

24. Find the radius of curvature of: 

The circle p = a sin 0; the lemniscate p 2 = a 2 cos 29; the logarithmic 
spiral p = c a0 ', the trisectrix p = 2a cos — a. 

25. If R is the radius of curvature of f(x, y) = o, show that 

R= (dx 2 + <fr»)* 

d 2 y dx — dy d 2 x % 
regardless of the independent variable. 

Differentiate the equation of the circle of curvature, 
R 2 = (x - af + (y- bf. 
o = (x — a)dx -J- (y — b)dy, 
o = dx 2 -f (x — a)</*# + ^' 2 -f (y — b)d 2 y. 
The elimination of x — a and y — b gives the result. 



CHAPTER XIII. 
ENVELOPES. 

102. Iff(x, y) = o is the equation of a certain line containing a 
constant a, then we can implicitly indicate that the position of this 
curve depends on the value of a by including it in the functional 
symbol, thus : 

A x > y> a ) = °- 

If we change a by substituting for it another number a l , we get 
another curve, 

A x > y> a i) = °> 

which will, in general, intersect the first curve. 

The arbitrary constant a in f(x,y, a) = o is called a parameter. 
All the curves obtained by assigning different values to a are said to 
belong to the same family of curves, of which a is the variable 
parameter. Thus 

f(x,y, a) =o (i) 

is the equation of a family of curves when we regard a as a variable, and 
any curve obtained by assigning a particular value to a is a particular 
member of that family. 

Thus, in the figure, let the curves i, 2, 3, ... be the particular 
curves of the family (1), obtained by assigning to a the particular 
values a x , a 2 , . . . taken in order. 




Two curves of this family are said to be consecutive when they 
correspond to consecutive values of a. The sequence of curves corre- 
sponding toff 1( ff 2 , . . . , as drawn in the figure, intersect in points 
A, B, C, . . . 

138 



Art. 103.] 



ENVELOPES. 



139 



Illustrations. 

The arbitrary constant or parameter being a : 

(</). y — ntx -f- <-* is the family of parallel straight lines sloped m to the axis 
ot .v. Consecutive members of this family do not intersect in the finite part of the 
plane. 

(/;). y — ax -f- b is the family of straight lines passing through the point o, b. 

(c). x cos a -j- y sin a = / is the family of straight lines tangent to the circle 
x*+y*=f*. 

(d). y — ax -\- b/a is a family of straight lines tangent to a parabola y' 1 = \bx, 
and 

y = ax — 2ba — ba 3 

is the family of normals to the same curve. 

(e). (x — a)- -\- (y — b) 1 = a 2 is the family of circles with center a, b and 
variable radii. The curves of the family do not intersect. 

(/). X s -j- )'' — 2 & x + r ~ = o is the family of circles with radius r having 
their centers on Ox. Two curves of the family do intersect, provided we take 
their centers near enough together. 



103. The Envelope of a Family of Curves. — If 

f{x,y, a) = 



(0 
(2) 



are two curves of the same family which intersect at a point x, y, let 
us seek to determine the limiting position of the point of intersection 
x,y when a 1 ( — )a. When a 1 ( = )a all points on curve (2) converge 
to corresponding points on (1), and in the limit curve (2) passes over 
into curve (1) and they have an infinite number of points in common. 
Therefore the attempt to determine the limiting position of the point 
x, y of intersection of (1) and (2), by solving (1) and (2) for the 
coordinates and then making a Y ( = )ct, leads to indeterminate forms. 

We shall proceed to find the limit to which converges the point 
x,y of intersection of (1) and (2), 
by finding a third line which also 
passes through their intersection, and 
which does not coincide with (1) 
when <t 1 ( = )o'. 

Assign to x and y the numbers a 
and b, the coordinates of the inter- 
section of (1) and (2), and let a be 
a variable number. Then _/"(#, b, a) 
is a function of the single variable a, 
and we have, by the law of the mean, 

/(a, b, a x ) -/(a, b, a) = (^ - 

where jjl is some number between ct 1 and a. 
But, a, b being on (1) and (2), we have 




Fig. 24. 



(3) 



Therefore 



/(a, b, a x ) = o and /(a, b, a) = o. 



(4) 



140 APPLICATIONS TO GEOMETRY. [Ch. XIII. 

For the particular value ju assigned in (3) we tiave/l(x 9 y, fj) — o 
as the equation to some curve passing through the intersection of (1) 
and (2), in virtue of equation (4). We do not know the number yu 
in (3) and (4), since all we know about it is that it lies between a 
and a . 

But, whatever be the number ju satisfying (4), we know that the 
curve 

/;(.v, y, ja) = o (5) 

passes through the intersection of (1) and (2). Now, when a x ( = )a, 
then ju(=)a. If, therefore, when or 1 ( = )o', the two curves (1) and 
(2) intersect in a point which converges to a fixed point as a limit, 
then (5) becomes 

/L(*> y> a ) = °> ( 6 ) 

the equation to a curve which passes through the limit of the inter- 
section of (1) and (2) as (2) converges to (1). Moreover, (6), being 
a curve distinct from (1), has in general a definite intersection with (1). 
If, between the equations 

f(x,y, a) = 0, (1) 

f*(x, y, a) = o, (6) 

the variable parameter a be eliminated, we obtain the locus 

E(x y y) = o (7) 

of all points in which the consecutive curves of the family fix, y, a) 
= o intersect as a varies continuously. 

The curve (7) is called the envelope* of the family (1). 

Illustration of the Envelope. 

As the parameter oc varies continuously, the curve f(x, y, a) = o sweeps over 
or generates a certain portion of the surface of the plane xOy, and leaves unswept 
a certain portion. The envelope may be regarded as the line which is the bound- 
ary between these two portions of the plane xOy. 

104. The envelope, E{x t y) — o, is tangent to each member of 
the family f{x, y, a) = o which it envelops. 

We are not prepared to give a rigorous proof of this statement 
now. This prouf requires a knowledge of functions of several 
variables. We can, however, give a geometrical picture which will 
illustrate the general truth of the statement. For this proof see 
§ 227. 

Let (A), (B), (C) be three contiguous curves of the family, (A) 

* Strictly speaking, the equation of the envelope is the equation gotten by 
equating to o that factor of E(x, v) which occurs only once in £(x, y). See 
Chapter XXXIX. 



Art. 105.] 



EXVELOrES. 



141 



and (C) intersecting the fixed curve (B) in points P and Q re- 
spectively. When (A) and (C) converge to coincidence with (B), 




(A) (B) (Q) 

Fig. 25. 



the points P and Q converge to each other and to two coincident 
points on the envelope. The straight line PQ converges to a common 
tangent to (B) and the envelope. 



EXAMPLES. 

The variable parameter being a, find the envelopes of the following curve 
families: 

1. x cos (x. -f- y sin a. — p = o = f(x, y, a). 

f' a — — x sin a -f- j cos a. Square and add. Hence 
jc* 2 -j- jy 2 = / 2 , a circle with radius /. 

2. Envelope the family _/ = y — ax — b/a = o. 

yX = — x + ^/« 2 - . *. a = \/b/x. Hence j 2 = 4&tr. 

3. Envelope the family f = y — ax -|- 2^0: -f £a: 3 . 

/^ = - x + 2£ + 3<$a 2 . .-. a 2 = (jc - 2^)/3^. Hence 
27jk 2 ^ = 4(x — lb f. 

4. Find the envelope of (x cos a) fa -(- (_y sin a:)/^ = 1 . 

5. Find the envelopes of y = ax -j- |/a 2 a 2 ± £ 2 . Er 2 /a 2 ± jj/ 2 /£ 2 = 1.] 

6. Envelope the family ;c 2 -f- _>' 2 — 2a;c = r 2 . 

105. Envelopes when there are Two Connected Parameters. 

Let (p(x, y, a, ft) = o (1) 

be the equation to a curve, involving two arbitrary parameters a and 
fi which are related by the condition 

4>(a, fi) = o. ( 2 ) 

I. When we can solve (2) with respect to a or /? and substitute 
in (1), we reduce that equation to that of a family with one parameter. 
The envelope is then found as before. 



142 



APPLICATIONS TO GEOMETRY. 



[Ch. XIII. 



EXAMPLE. 




Find the envelope of the ellipses 



a 2 ^ /j 2 



(i) 



when a -\- ft = c. 

We have /? = c — a. Therefore 



a 2 {c — af 



Differentiating with respect to or, and 
solving for a, 



ex 



■j+j 



and /3= 



cy< 



J + f 



Fig. 26. 



which substituted in (1) give 
x* -f y% = A 



II. Otherwise, when it is inconvenient to solve (2), it is generally 
simpler to proceed as follows : 

~Letx,j> be constant, and differentiate (1) and (2) with respect to 
any one of the parameters, say /3. Eliminate a, fi and a' = da/d/3, 
between the four equations. 

(p(x,y, a, (5) = o, (1) 

(t>lt(x,y, a, (3) = o, (2) 

$(a, fS) = o, (3) 

#(*, p) = o. ( 4 ) 

The result is the envelope E{x, y) = o. 



For example, take the same question proposed in I. 
We have for'(i), (2), (3), (4), 



f , 7 
a* r (S 1 



x- 
—JOC 

or 



— o, 



: 2 

a + fizzc, 

a' -f 1 = o. 
The elimination gives the same result as before. 



(2) 

(3) 

(4) 



EXERCISES. 

1. Find the envelope of a straight line of given constant length, whose ends 
move on fixed rectangular axes. [jr 1 -|- _y 3 — ^r 5 .] 

2. Find the envelope of the ellipses 

x? r 2 _ 
a 2 + & ~ ' 
when the area is constant. [2^ry = c 2 .] 



A.RT. 105.] ENVELOPES. 143 

3. Find the envelope of a straight line when the sum of its intercepts is con- 
stant, [x- _[_ yh — A] 

4. One angle of a triangle is fixed; find the envelope of the opposite side when 
the area is given. [Hyperbola. J 

5. Find the envelope of x/a -\- y/(3 = 1 when a" 1 -f fi m = c>". 

r jn_ jn_ rn_ "1 

|_X'" + i -^ y/rn+i = fW+i.J 

6. Show that the envelope of x/l -\- y/m = I, where l/a -\- m/b = 1 is the 
parabola (x/a)- -f (y/6) h = 1. 

7. From a point Pon the hypothenuse of a right-angled triangle, perpendiculars 
PM. /Ware drawn to the sides; find the envelope of the line MN. 

8. Find the envelope of the circles on the central radii of an ellipse as diameters. 

9. Find the envelope oiy = lax -f- t* 4 . [16/ 3 -\- 27a- 4 = o. ] 

10. Find the envelope of the parabola y 2 = a(x — a). [4y 2 = x-.] 

11. Find the envelope of a series of circles whose centers are on Ox and radii 
proportional to their distances from O. 

12. The envelope of the lines x cos 3a -|- y sin 3a = a(cos ia)% is the 
lemniscate (x 2 -|- y 2 ) 2 = a\x 2 — y 2 ). 

13. Find the envelope of the circles whose diameters are the double ordinates 
of the parabola j 2 = ^ax. [y 2 = \a{a -j- x).] 

14. Find the envelope of the circles passing through the origin, whose centers 
are on y 2 = \ax. [(x -\- 2a)y 2 -\- x' 6 = o.J 

15. Find the envelope of x 2 /a 2 -j- J 2 //? 2 = 1, when a 2 -j- fi l = & 2 - 

[(x ±yf =&.] 

16. Circles through O with centers on x 2 — y 2 = a 1 are enveloped by the 
lemniscate (x 2 -f- y 2 ) 2 = ^a 2 {x 2 — y 2 ). 

17. Show that the envelope of 

La 2 + zMa -f N= o, 

in which L, M, N are functions of x and y, and a is a variable parameter, is 
LN= M 2 . 

18. In Ex. 17 show that if Z, M, A 7 " are linear functions of x and y, the envelope 
is a conic tangent to L = o, N = o and having 7J/ = o for chord of contact. 

Differentiate LN — M 2 = o with respect to x, 

... L'N-{- N'L = 2MM'. 

At the intersection of L = o and J/ = o we have L' N = o; and since there 
/V ?£ o, we have Z' = o. The D x y from this is the slope of the tangent to the 
envelope. Hence U = o is the tangent at the intersection of Z = M = o to the 
envelope, etc. 



CHAPTER XIV. 
INVOLUTE AND EVOLUTE. 

106. Definition. — When the point of contact, P, of the circle of 
curvature of a given curve moves along the curve, the center of curva- 
ture, C, describes a curve called the evohile of the given curve. 

The evolute of a given curve is the locus of its center of curva- 
ture. The given curve is called an involute of the evolute. 

107. There are two common methods of finding the evolute of a 
given curve. 

I. If <p(x, y) = o is the equation of the given curve, and a, (3 are 
the coordinates of the center of curvature, then we have, § 100, (2), 

a-x=-/^-, fi-y = -^-. (!) 

If we eliminate x and y from these two equations and the equation 
to the curve, <p{x, y) = o, we leave a and fi tied up with constants 
in the equation to the evolute. 

Eliminations are, in general, difficult and no general rule can be 
given for effecting them. Another method of finding the evolute will 
be given in II, which frequently simplifies the problem. 

EXAMPLES. 

1. Find the evolute of the parabola y' 1 = \px. 

1 1 1 3 

We have y' — /\r 2 ; y" — — \p x -. Hence 

a — x = 2(x +/), ft - y — - 2(/~*.r ? +/***). 

. •. a — ip -f ix, fi — — 2/ _ '- , .r-\ 

Eliminating x, we have for the equation to the evolute (§ 112, Ex. 17, Fig. 44) 

4(« - 2p? = 27//S-. 

2. To find the evolute of the ellipse x 2 /a 2 -\- y 2 /b 2 = 1. 
We can differentiate directly, or use the eccentric angle 

x = a cos (p, y = b sin (p, and find 
/ ~ - b 2 x/a 2 y, y" = - $*/<*y. 



a 2 — b 2 


a 2 - />-' 


*« -' 


aa) 1 -f (/'/3)° 3 = (« a - 


b 2 )\ 


144 





Hence (aaY -f (V'/3) 3 = (a 2 - b 2 )\ (Fig. 43.) 



Art. ioS.] INVOLUTE AND EVOLUTE. 145 

II. The evolute of a given curve_/(.v, y) = o is the envelope of 
the normals to the curve. 

The equation to the normal to/" = o at x,y is 

X-x+(V-y)y' = o. (1) 

But ;■ and y' are functions of x, from the equation f= o to the 
curve. Therefore x is a parameter in (1), by varying which we 
get the system or family of normals. Hence the required locus is to 
be found by differentiating (1) with respect to x, keeping X, incon- 
stant. Thus 

_i + (jr-jy-y» = o. (2) 

Eliminating x between (1) and (2), we have 

1 -4- v' 2 1 4- v' 2 

Y-y = \f and X-x=-y' \f- , 

y y 

in which X and JTa.re the coordinates of the center of curvature, 
S 100, (2). This proves the statement. 

EXAMPLES. 



The equation to the normal is 

y = ax — 2pa — pa 3 . (1) 

. •. o = x — ip — spa 2 . 

\ ip ) • 

which substituted in (1) gives as before in I, \{x — 2fi) 3 = i-jpy" 1 . 
2. Find the evolute of the ellipse x 2 /a 2 -f- y 2 /b 2 = 1. 
Taking the equation to the normal 

ax see a — by esc a = d l — b 2 . 
ax sec a tan a -f- by esc a cot a = o. 

Hence tan a = — (by /ax) 3 , which leads to the same result as in I, 



(axf + (byf = (a 2 - b 2 f. 

108. The normal to a curve is a tangent to the evolute. 

Let (x-a)*+(y-fi)* = lP (1) 

be the equation of the circle of curvature at x, y. Then, letting x f y 
vary on the circle, R remaining constant, we have, on differentiation 
with respect to x t 

x — a -f O — fi\\>' = o, (2) 

i+y+^-w^o. (3) 

Now let x, v vary along the curve, R being variable. The num- 
bers a and /? are also functions of x. Differentiate (2), which is the 
equation to the normal to the curve at x,y, with respect to x. 

• •• 1 +y* + <y- my - <*' - PY = °, (4) 



146 APPLICATIONS TO GEOMETRY. [Ch. XIV. 

Subtract (4) from (3). 

da d/3 dy 
**• dx~ + dxdx~ = ° f 
d6 dx 

or ■£- = - TT» 

da dy 

which proves the statement. 

EXERCISES. 

1. Find the centre of curvature of y' 6 — d l x. 

_ a * + I 5; /4 R _ ai )' - 9J>' 5 
CX ~ 6a*y ' ' ~ 2tf ' 

These equations are the equations of the evolute. a and being expressed in 
terms of y, a third variable. 

2. Find the coordinates of the center of curvature of the catenary, Fig. 38, 

( ~ --) 

y = la \e a -\- e a ) . 

y 



a — x — — \/y' 2 — a 2 , ft = iv. 
a 

3. Find the center of curvature and the evolute of the hypocycloid, 

a = x + 3 *V, ft =y + 3* § A (<* + pf + (a - /J) 1 = 2 J. 

4. In the equilateral hyperbola ixy = a 2 , 

(a -f /i) 1 - (a - /J) S = 2a 1 . 

5. In the parabola x* -\- y* = a , a -\- /3 = 3(.* -J- ^)« 



CHAPTER XV. 

EXAMPLES OF CURVE TRACING. 

109. Until functions of two variables have been studied we are 
not in position to consider the general problem of curve tracing in 
the most effective manner. Nevertheless it will be advantageous to 
apply the properties of curves which have been developed for func- 
tions of one variable to rinding the forms of a few simple curves, 
whose figures will be useful in the sequel, before we study functions 
of more than one variable. 

no. Principal Elements of a Curve at a Point. — We collect 
here for handy reference the principal elements of a curve at a point, 
as deduced in the preceding pages. The notations are the same as 
there used. 

I. Rectangular Coordinates. D x y = y', D x y = v" . 

1. Equation of the tangent: 

{Y-y) = (X-xy. 

2. Equation of the normal : 

{ F- y y=-{x-.x). 

3. Subtangent and subnormal : 

s, = \y'~\ s n =, y y. 

4. Tangent-length and normal-length : 

/ =yVi -\-y'~ 2 , n =yVi +j>' 2 . 

5. Tangent intercepts on the axes: 

Xi = x -yy f -\ Fi=y- xy' . 

6. Perpendicular from origin on the tangent: 

7. Radius of curvature: 

■i+y»|i 



R = 



y" 



\. Coordinates of center of curvature: 



y 

147 



14S APPLICATIONS TO GEOMETRY. . [Ch. XV. 

II. Polar Coordinates. D e p = p' ', Dip = p" . up = 1. 

1. Angle between the tangent and radius vector: 

tan ib = —p. 
P 

2. Angle between the tangent and the initial line: 

p -j- p' tan 6 

tan = —. 7l . 

^ p — p tan 6 

3. Subtangent and subnormal : 

^ ~ ^ - " 5P 6 * - p - ^' 

4. Tangent-length and normal-length: 

5. Perpendicular from the origin on the tangent: 

p 2 1 



iV + p' 2 / 

6. Radius of curvature 



= , — = «2 + «' 2 . 



7v J 



(P 2 + P' 2 f- 



P 2 + 2P^ - pp' 
( w 2 _|_ w /2)| 



in. Explicit One-valued Functions. — If the equation to a 
curve can be solved with respect to the ordinate or the abscissa so as 
to give 

y = <p(x) or x — ip(y) 

as its equation, in which either (p(x) or tp(y) is a one-valued func- 
tion, or if more than one-valued the branches can be separated, we 
have the simplest class of curves for tracing. 

Given any value of the variable, we can compute the value of the 
function. We thus obtain the coordinates of a point on the curve. 
By finding the first and second derivatives, y' , y", we can compute 
all the elements of the curve at this point, y' gives the direction 
andjy" the curvature at the point. 

A regular method of procedure for tracing a curve is: 

1. Examine the equation for symmetry. 

If the equation is unchanged when the sign of y is changed, the 
curve is symmetrical with respect to Ox. 

If the equation is unchanged when the sign of x is changed, the 
curve is symmetrical with respect to Oy. 

If the equation is unchanged when the signs of x and y are 
changed, the curve is symmetrical with respect to the origin which 
is a center of the curve. 



Au 



EXAMPLES OF CURVE TRACING. 



149 



If the equation is unchanged when x and y are interchanged, the 
curve is symmetrical with respect to the line y = x. 

If the equation is unchanged when x and y are interchanged and 
the signs of both x and y changed, the curve is symmetrical with 
respect to x -\- y = o. 

2. Examine for important points. 

These are: the origin, the points where the curve cuts the axes, 
maximum and minimum points, and points of inflexion. 

If x = o, y = o satisfy the equation, the curve passes through 
the origin. Put x = o and solve for y to find the intercepts on Oy; 
put y = o and solve for x to find the intercepts on Ox. 

Find the maximum and minimum and inflexion points by the 
regular methods of the text. 

3. Determine the asymptotes, if any. 

4. Compute a sufficient number of points on the curve to give a 
fair idea of the locus, and sketch the curve through the points. 

(In the following examples all details, omitted in the hints, must be supplied.) 



EXAMPLES. 

1. Trace the common parabola y = x 2 . The curve is symmetrical with respect 
to Oy. It passes through O and cuts neither axis else- 
where. Since y' = 2x is o at O, Ox is tangent. 
Also, y' is positive as x continually increases from o, 
and y, the ordinate, continually increases. Since 
y" = 2 is always -j-, the curve is everywhere convex. 
Investigation shows that the curve has no asymptote. 
The form of the curve is as in the figure. (Fig. 27.) 

2. Trace y — 2x 2 — t,x -j- 4. 

y' - \x — 3, y" - 4. 
The curve is always convex. It has a minimum 
ordinate, y = 2|, at x = |. Its slope ± according as 
x < 3. It cuts Oy at y = -j- 4, and neither axis else- 
where. It is symmetrical with respect to the line x = 5. The curve is the com- 
mon parabola. (Fig. 28.) 

V 





y 







N X 







Fig. 29. 

3. Trace x = 3 — zy — $y 2 . Here x is a one-valued function of y. 
D y x = — 2 — 6y, DyX = — 6. x is a maximum at y = — \, when x = 3^. 
The curve cuts Ox at y = o, x = 3, and Oy at y = -f- 0.78, ^ =■ — 1.44. It is 
everywhere concave to Oy. x continually diminishes from its maximum value, 



APPLICATIONS TO GEOMETRY. 



[Ch. XV. 



and the curve has no asymptote at a finite distance. It is as before the common 

parabola. (Fig. 29.) 

4. Trace the cubical parabola y = x 3 . Here 
y = 3-v' 2 , y" = 6x. The curve passes through O. It 
is symmetrical with respect to O, since the equation is 
unchanged when the signs of x and y are changed. The 
ordinate is -\- when x is -)-, and y is — when x is — . 
The curve lies in the first and third quadrants. In the 
first quadrant it is everywhere convex, in the third every- 
where concave to Ox. It changes its curvature at the 
origin where there is concavo-convex inflexion. There 
are no asymptotes and the absolute value of y is 00 when 
that of x is 00 . (Fig. 30.) (Read foot-note, p. 164.) 

5. Trace the semi-cubical parabola y 2 = x 3 . Ox is 
an axis of symmetry. The origin is on the curve. When 
x is — , y is imaginary and the curve does not exist in the 
plane to the left of Oy. x = y* is a one-valued function 

shows the slope 00 at O with respect to Oy, and this slope is 
Dlx = - £v— a is always negative, or the curve is con- 
y become 00 together. (Fig 31.) 




Fig. 30. 
of y. D y x = \y 
± fory ± respectively. 
cave to Oy. There are no asymptotes and x, 





Fig. 31. 



Fig. 32. 



6. Trace the logarithmic curve y = log x. We adopt the convention that the 
logarithm of a negative number is imaginary. Then the curve does not exist as a 
continuous function to the left of Oy. The ordinate is negative and infinite for 
x = o, positive and infinite for x = -f- 00 , and is o when x = 1 where it cuts the 
axis Ox. The derivative y' = i/x is infinite for x = o, which line is an asymp- 
tote, y' is always positive and decreases as x increases. The ordinate continually 
increases, y" = — x~ 2 is always — , hence 
the curve is everywhere concave and as in 
the figure. (Fig. 32.) 

7. Trace the exponential curve y = e x . 
y is always -}-, by convention e x is -f. 
y = -f- 00 when x = -)- 00 ; y(=)o when 
x = — 00 . Also y = y" = e x . The curve 
is always convex and increasing, and since. 
y'( = )o when x — — 00 , Ox is an asymptote. 
When x = o, y = 1, where the curve cuts 
Oy. If we agree with some authors that y 
has negative values for x = (2« -(- i)/2tn. 
tn and n being integers, then there will be a corresponding series of points 
representing the function lying below Ox on a curve represented by a dotted 




Art. hi.] EXAMPLES OF CURVE TRACING. 



5 1 



line symmetrical with that above Ox. The exponential curve, however, i> con- 
ventionally taken to be the locus of the equation 

, a v .t 
v = i + x + - + -- + . . . = e*. 

2. j. 

The curve y = e* is identically the same as the curve in Ex. 6 if we inter- 
change x and y. (Fig. 33.) 

8. Trace the probability curve y = c—* 1 . The ordinate is always -(-; it has a 
maximum at o, 1 ; it is o when x is ± 00 . There is a concavo-convex inflexion at 




- 1/V2. Ox is an 
Show that the curve 



x = -f- i/V 2 and a convexo-concave inflexion at x = 
asymptote in both directions, and Oy an axis of symmetry, 
is as in the figure. (Fig. 34.) 

9. Trace the cissoid of Diodes, (2a — y)x 2 = y z . The curve has Oy as an 
axis of symmetry, and passes through O, and cuts the axes nowhere else. Since 




Fig. 35. 

y* _^_ x 2 y = 2<ix 2 , y cannot be negative if a is positive. We find that y = 2a is 
an asymptote in both directions, since x = ± 00 when y = 2a. 

Again, corresponding to an assigned y, there are only two equal and opposite 
values of x. Therefore, for an assigned x, there is only one value of y. Also, 



IX 



2a — y 



3/ z + x l 

is ± according as x is ± . The curve is decreasing for x negative and increasing 
for x positive. To find;/ at the origin, the above value of y' is indeterminate. 
But we have directly from the equation to the curve 



/©-/=?*- 



which is the slope of the curve at O. Therefore Oy is tangent to the curve. The 
origin, like that in Ex. 5, is a singular point which we call a cusp. By plotting a 
sufficient number of points, we find the curve to have the form as drawn in the 
figure. (Fig. 35.) 



15 2 APPLICATIONS TO GEOMETRY. [Ch. XV. 

10. Trace the witch of Agnesi, y = Sa 3 /(x 2 -f 4a' 2 ). The ordinate is always 




-\-, and has a maximum y = 2a, at x = o. Oy is an axis of symmetry, and Ox 
an asymptote. There are inflexions at x = ± 2a/ 4/3. (Fig. 36.) 

11. Trace the cubic a 2 y = ^x 3 — ax 2 -\- 2a 3 , in which a is positive. There is 
a maximum y = 2a at x = o ; and a minimum y — |a. at jt = 2a. An inflexion 
occurs at x = a. For x < a the curve is concave, and for x > a convex. There 
are no asymptotes. The curve crosses Ox between x = — a and x = — 2a. 
Also, y = ± 00 when x = ± co . (Fig. 37.) 





Fie. 37. Fig. 38. 

(- --\ 

12. Trace the catenary, y = %a\e a -\- e a) , in which a is a positive constant. 

The curve is the form of a heavy flexible inextensible chain hung by its ends. 
The ordinate y is a minimum and equal to a when x = o, and is -f- for all values 
of x. The curve is convex everywhere, y — -\- 00 when x = ± 00 , and there are 
no asymptotes. The slope continually increases with x. (Fig. 38.) 

13. Trace the cubical parabola x 1 = y 2 (y — a), where a is positive. 

Since x = ± y \/y — a, 

the point o, o is on the curve. But no other point 
in the neighborhood of the origin is on the curve y 
since for such values of y, x is imaginary. The 
origin is therefore a remarkable point, it is an iso- 
lated point of the curve, and such points are called 
conjugate points. For each value of y greater than 
a there are two equal and opposite values of x. 
The curve is symmetrical with respect to Oy. The 
ordinate y is a minimum at x = o. where the tan- 
gent is horizontal. y" = O gives inflexions at 

_ 4 _ 4 3 

— ~- " 2 which for x A- is 






Fig. 39. 
convexo-concave and for x — 



y = -\- so for x 



± 00. 



is concavo-convex. 
(Fig- 39-) 



3^3 
There is no asymptote, and 



Art. HI.] EXAMPLES OF CURVE TRACING. 1 5 J 

14. Trace y = (x 2 — 1)- The curve lies above Ox and has Oy for an axis 

y 




Fig. 40. 

of symmetry, y has a maximum at x = o, and minima at x = ± 1 . There are 
inflexions at x = ± 1/ \/ r $. 

The infinite branches have no asymptote. (Fig. 40.) 



15. Trace the curve y 



-(•+*) 




The ordinate has the limit e when 
x = ± & . This is the important 
limit on which differentiation was 
founded, y has the limit 1 when x = o 
and continually increases with x. For 
— 1 < x < o the curve does not exist. 
The point o, 1 is what is called a stop 
point, the branch ending abruptly 
there. For x < — I, and converging 
to — 1, y is greater than e and is 00 . 
As x decreases to — so , the curve 
decreases continually and becomes 
asymptotic toj = e. (Fig. 41.) 



EXERCISES. 

1. Trace the curves y = sin x, y =. cos x. 

2. Trace y = tan x, y = cot x. 

3. Trace y = sec x } y = esc jr. 

4. Trace j = vers .r = 1 — cos x. 

5. Trace y = e x . 

6. Trace the curves 

xy = 1, (* - i)(j - 2) = 3, y(x - i)(x - 2) = 1. 

7. Trace the curve y(x — i)(x — 2) = (x — 3)^ — 4). 

8. Trace y(a 2 -f- x 2 ) = a 2 (a — x). 

9. Trace x\y — a) = a % — xy 2 . 

10. Trace a 2 x = _j/(j: — a) 2 . 

11. Trace j 3 = jr 2 (2^ — ^r). 

12. Trace (x 2 4- 4) = jv^ 3 . 



J54 



APPLICATIONS TO GEOMETRY. 



[Ch. XV. 



Tt(a — x) 



la 



a)*(x 



«f- 



13. Trace 3^r(i — x)y = I — $x. 

14. Trace the quadratrix y = x tan 

15. Trace the curve y = sin {it sin x). 

16. Trace y = (2x 

112. Implicit Functions. — In general, when the equation to a 
curve is given in the implicit form f(x, y) = o, and we cannot solve 
for either variable, the investigation requires more advanced treatment 
than we are prepared to give here. This subject will be taken up 
again in Book II. The ordinatesto such curves are, in general, several- 
valued functions of the variable. 

We give here simple examples of important curves. The student 
will do well to study the hints given in tracing such curves. 
15. Trace the hypocycloid of four cusps, 

x * + y — a ^- 

The curve is symmetrical with respect to O, Ox, and Oy. There are two 
equal and opposite values of y to each x, and two of x to each y, for either variable 




Fig. 42. 

less than a. The curve does not exist for values of x or y greater than a. We have 
in the first quadrant 

' — IS 

and the curve is tangent to Ox at x = a, and to Or at .v = o, y = a. y" being 
positive in the first quadrant, the curve is convex at any point on it. The curve 
is sometimes called the asteroid. It is the locus of a fixed point on the circumference 
of a circle as that circle rolls inside the circumference of another circle whose radius 
is four times that of the rolling circle. (Fig. 42.) 

16. Trace the evolute of the ellipse 



(axf + (/>y)* = (a* - &*) 



in the same way as above. (Fig. 43.) 



Ar 



EXAMPLES OF CURVE TRACING. 



55 



Show by inspection that four normals can be drawn to the ellipse from any point 
inside the evolute. 

From what points can I, 2, or 3 normals be drawn? 





Fig. 43. 



Fig. 45. 



17. Trace the parabola y 2 = \px, and its evolute, \{x — ipf = 27 py*. Show 
that the curves are as drawn. Find the angle at which they intersect. Show from 
which points in the plane can be drawn I, 2, or 3 normals to the parabola. (Fig. 44.) 

18. Trace the curve ( y — x' 1 ) 2 = x 5 . 

.-. y = x*{i ± **). 
There are two branches, 

y = x 2 (l + x i ), y = x 2 (l _**). 

The first continually increases as x increases from o. The second increases, 
attains a maximum, and then descends indefinitely, crossing Ox at x = I. Both 
branches are tangent to Ox at O since 



is o when x = o. The curve does not exist in the plane to the left of Oy. Ex- 
amine for asymptotes. Find the inflexion and the maximum ordinate. The origin 
is a singular point called a cusp of the second species. (Fig. 45.) 

19. Trace in the same way the curve 

x 4 — lax' 1 } 1 — axy 2 -j- a' 2 / 2 = O. 



20. Trace the curve 



1/2 



TZ (X -f- I)X 2 . 



y is a 



two- valued function of 



y z= ± x \'x + I. 

Ox is an axis of symmetry. 

The curve passes through the origin in two branches, 

y = + x \/x~+~\, y = - x \/x + r. 

The curve does not exist in the plane to the left of x 




156 



APPLICATIONS TO GEOMETRY. 



[Ch. XV. 



and o the ordinate is finite, having a maximum and a minimum. We have for the 
slopes of the two branches passing through O at x = o. 



*W 



*(=)0 



V* + 



As x increases positively, y increases without limit in absolute value. Are there 
asymptotes? (Eig. 46.) 

The point in which two branches of the same curve cross each other, having 
two distinct tangents there, is called a node. In this curve the origin is a node. 

21. Trace the curve (bx — cy) 1 = (x — a) 5 . 

Clearly, x = a, y = ab/c, 

is on the curve. But these values make the deriva- 
tive y' indeterminate. Differentiate the equation 
twice. 
.-. (b — cy'f — (bx — cy)cy" — \o(x — af — o, 
and at the point x — a, y = ab/c, 




(b - cy'f 



Fig. 47- 



b/c. Since y is imaginary when x < en 

b 

y = 



- x ± — \/(x - af 



gives y 

and 

the curve is as in the figure. The point a, ab/c, is a cusp of \hejirst species. (Fig. 47. ) 
22. Trace the curve 4y~ = 4.x 3 -\- i2x 2 -(- gx . 

y 




Fig. 48. 
23. Trace the lemniscate, 



shows that y cannot be greater than x and only equal to^r when they are both o at 
the origin. The curve is symmetrical with respect to O, Ox, Oy. Also, 



i - (;)*' 



and since y -^ x, we have, when x = O, y = o, 



£■ 



± 1, 



(See 111. (2), § 79.) 



which are the slopes of the two branches of the curve passing through the origin. 
Again, 

x a* - 2(x*+y*) 

y 'a 2 -f 2(a- 2 + j' 2 ) ' 



Art. 113.] EXAMPLES OF CURVE TRACING. 157 

In the first quadrant y' is -f from x = o to the point determined by 

2(-v- - y 2 ) = a\ 4(-<' 2 - )' 2 ) = a\ 
where it changes ^ign. giving y a maximum, and y' decreases until y' = 00 at 
,r = a, v = o. 

Being symmetrical with respect to the axes the curve is as in the figure. No 
part of the curve exists for x > a, since the equation is of the fourth degree and a 
straight line cannot cut the curve in more than four points. 

Put j' = mx, and plot points on the curve by assigning different values to m. 
Thus, in terms of the third variable m, we have 

* = ± a \ TW -> y=± am iL_, (Fig. 48.) 

113. General Considerations in Tracing Algebraic Curves. — 

The equation of any algebraic curve when rationalized is of the form 
of a polynomial of the «th degree in x and y. It can always be 
written 

o = « + «x + -..+««= V (I) 

where u is the constant term (independent of x and y), u x , u 2% 
etc., are homogeneous functions or polynomials in x, y of respective 
degrees 1, 2, etc. 

If u = o, the origin is a point on the curve. 

(1). To find the tangent at the origin when u x 7^ o. 

When « = o, the line^y = mx intersects the curve at O. 

Substitute mx for y in the equation to the curve. Then, if 
u x = px -\- £>', the equation (1) becomes 

(P + m)x + T 2 + . . . = o, (2) 

where the terms T 2 , etc., contain higher powers of x than the first. 
Divide the equation (2) by x, which factor accounts for one o root. 
Then let x = o, and (2) becomes 

p -f- qm = o, or m = — p/q. 

This value of m is the slope of the curve at the origin, since now 
the line y = mx cuts the curve in two coincident points at the origin, 
and 

u x = px + qy = o 

is the equation of the tangent at the origin. 

If u = o, u x = o, and u 2 = rx z -\- sxy -f- ty 2 . 

Then, as before, put mx iovy and the equation becomes 

(r + sm + tm l )x 2 + T 3 + . . . = o, (3) 

where the terms T 3 , etc., contain higher powers of x than 2. 

Divide by x 2 , which accounts for two zero roots of (3); in the 
result put x = o. 

. • . tm 2 -\- sm -\- r = o (4) 

is a quadratic giving two values of m, the two slopes of the curve 
at 0. The equation to the two tangents at O is 

u = rx 2 -f- -m^ + ^>' 2 = o. 



158 APPLICATIONS TO GEOMETRY. [Ch. XV. 

These are real and different, real and coincident, or imaginary, 
according as the roots of the quadratic (4) in ??i are real and unequal, 
equal, or imaginary. The origin being a double point called a node, 
cusp, or conjugate point accordingly. 

In like manner if also u 2 = o, the equation of the three tangents 
at O is 

u s — o, 
and the origin is a triple point. 

Hence, when the origin is on the curve, the homogeneous part 
of the equation of lowest degree equated to o is the equation of the 
tangents at O. 

Further discussion of singular points and method of tracing the 
curve at a singular point will be given in Book II. 

(2). A straight line cannot meet a curve of the ;zth degree in more 
than n points. For, if we put mx -{- b for y in U = o, we have an 
equation of the wth degree in x for finding the abscissae of the points 
of intersection oiy = mx -J- b and U = o. 

If now u r is the term of lowest degree in U, and we put mx for 
y in U, then x r is a factor and represents r roots equal to o. The 
linejy = mx cuts the curve £7= o, r times at the origin, and there- 
fore cannot cut it in more than n — r other points. This will fre- 
quently enable us to construct a curve by points, when otherwise the 
computations would be quite difficult. 

(3). Singular Points. A point through which two or more 
branches of a curve pass is called a singular point. Illustrations 
have been given of nodes, cusps, and conjugate points. 

At a singular point on a curve D x y is indeterminate. Points at 
which D x y is determinate and unique are called points of ordinary 
position, or ordinary points. 

To find a singular point on a curve cp(x, y) = o, differentiate 
with respect to x. The result will be 

M-^-Ny' — o, (1) 

where vl/and N are functions of x and y. At a singular point y' is 
indeterminate and Af= o, A r = o. Any pair of values of x, y satis- 
fying the equations 

= o, M— o, N — o 
is a singular point. If (1) be differentiated again, we have 
P + Qy' + Ry'* + Ny" = o. 

At the singular point N = o, leaving a quadratic in y' for deter- 
mining the slopes of the curve, if the point is a double point. If a 
triple point, another differentiation will give a cubic in y' for deter- 
mining the slopes, etc. 

If the curve has a singular point whose coordinates are a, /3, and 
we transform the origin to the singular point by writing x -j- a, 
y -\- fi for x and y in the equation to the curve, the construction of 
the curve will be simplified as in (1), (2). 



Art. ii 



EXAMPLES OF CTRYE TRACING. 



T 59 



EXAMPLES. 

24. Trace the lemniscate, Ex. 23. 

{x*+y*)* - a \x* -y*) = o. 

Here n 2 = x- — j'-' r= o is the equation to the tangents at o, or y = ± •*", as 
before in Ex. 23. 

Put r = w.r in the equation and compute a number of points. Clearly m 
cannot be greater than 1. 

25. Trace the folium of Descartes, 



x* 4- y* 



Zaxy = o. 




The equation of the tangents at the 
origin is $xy = 0, or x = o, y = o. We 
find that 

x + J + a = o 
is the only asymptote. Put j == mx, then 

"*tfW 3#W 2 

I-fw 3 ' -* I + ;// 3 ' 
jr, jj' are finite for o < m < -(- °° ■ Com- 
pute a number of points corresponding to 
assigned values of m. Observe that if we 
change m kito l/m, x and y interchange 
values. The curve is symmetrical with 
respect to the line y = x. In the first 
quadrant there is a loop, the farthest point 
from the origin being x =y = fa. Determine the maximum values of x and y for 
this loop. For negative values of m we construct the infinite branches above the 
asymptote, since y = mx cuts the curve before it does the asymptote. (Fig. 49.) 

26. Trace the curve (y — 2) 2 (x — 2)x = (x — i) 2 (x 2 — 2x — 3). 
Examining for singular points, we find 

y _ \ X\X — 2)7 7 — 2 

Therefore x = I, y = 2 is a singular point. Transform the origin to this 
point by writing x -f- I for x, y -f- 2 for/. 

Then the equation becomes 

y\o? - i) =-- x\* % - 4). 

Examining for asymptotes, we find the 
asymptotes x = ± I, y = ± x. The 
equation to the tangents at is y 2 = \x 2 , 
.'. y — ± 2x. When y = O; x = ± 2, 
^ = o. The curve is symmetrical with 
respect to Ox, Oy, and O. We need there- 
fore trace it only in the first quadrant, in 
order to draw the whole curve. 

The line y = mx cuts the curve in points 
whose coordinates are 



)7 7-2 




\± — m 2 Id. — m' 



These increase continually as m increases 

from o to 1, and the branch approaches the 

asymptote as drawn. The coordinates are 

imaginary for 1 < m < 2, and when m = 2: 

FlG. 50. x = O, y = O. As ;;z increases from 2 to 

-f- 00 , x and jj/ are real and increasing, and 

m — 00 gives jr = ± 1, y -= 00, the curve approaches the asymptote as drawn. 

The origin is an inflexional node. (Fig. 50.) 



6o 



APPLICATIONS TO GEOMETRY. 



[Ch. XV. 



27. Trace the curve (x -\- $)y 2 = x(x — i)(x — 2). 

28. Trace the curve a*y 2 = bx* -j- x b . 

29. Trace the dumb-bell a*f- = a-x* — x 6 . 

30. Show that ^-f/ = sax 2 )' 2 has the form given in Fig. 51. 





Fig. 51. 

31. Trace x 4 = {x 2 - y 2 )y. 

The lowest terms are of third degree. The origin is a triple point. The 
tangents there being y = o, y = ± x. Oy is an axis of symmetry. There are 
no asymptotes. The line y = mx cuts the curve in one point, besides the origin, 
whose coordinates are 

x = m{\ — m 2 ). y = m 2 (i — m 2 ). 

This shows that there are two loops, in the first and fourth octants, and infinite 
branches in the sixth and seventh octants. The curve is a double bow-knot and has 
no asymptotes. (Fig. 52.) 

* 




Fig. 53. Fig. 54. 

32. Trace the curves 

y' A — ax 2 — x*, y z = a 3 — x 3 , y 2 (x — a) = (x — b)x 2 . 

33. Trace the conchoid of Nicomedes, 

(*■ + V-){b - yf = ay*, when b =, <, > a. 

34. Trace the curves 

y — (x — i)(x - 2)(x - 3), a 2 x = y(b 2 + x 2 ), x* - y* + 2axy % = o. 

35. Show that x 2 y 2 -\- x* = a 2 (x 2 — y 2 ) consists of two loops and find the form 
of the curve. 

36. Show that the scarabeus 

4 ( X 2 + yi + 2 axf(x 2 + y 2 ) = b 2 (x 2 - y 2 ) 2 
has the form given in Fig. 53. 



Art. 114.J EXAMPLES OF CURVE TRACING. 161 

37. Show that the devil 

y* — x* -f~ ay 2 -f- &x 2 = o, where a — — 24, 3 = 25, 
has the figure given (Fig. 54). 

114. Tracing Polar Curves.— As in Cartesian coordinates, no fixed 

rule can be given for tracing these curves. The general directions 
are the same as before. The particular points are : 

(1). Compute values of p corresponding to assigned values of 6, or 
vice versa, according to convenience. Plot a sufficient number of 
points to give a fair idea of the general position of the curve. 

(2). Determine the asymptotes, by finding values of which make 
p = co for the directions of the asymptotes. Place the asymptote in 
position by evaluating the limit of p 2 D p 6 = — D U H, for the perpen- 
dicular distance of the asymptote from the origin, as previously 
directed. Examine for asymptotic points and circles. 

(3) . The direction of a polar curve at any computed point is given 
by tan tp = p/p'. 

(4). Examine for axes or points of symmetry. 

(5). Examine for maximum and minimum values of p and for points 
of inflexion. 

(6). Examine for periodicity. 

115. Inverse Curves.— If /(p, 6) = o is the polar equation to 
any curve, then /(p _I , 6) = o is the polar equation of the inverse 
curve.* We have been accustomed to put p _I = u, so that /"(a, 6) = o 
is the equation of the inverse curve. 

1. Show that if x, y are the rectangular coordinates of a point on a curve, the 
equation to the inverse curve is obtained by substituting 



x2 ~\~ y 2 ' x 2 -\- y' 2 
for x and y in the equation to the given curve. 

2. Show that the asymptotes of any curve are the tangents at the origin to 
the inverse curve. 

3. Show that a straight line inverts into a circle and conversely. Note the case 
when it passes through the origin. 

4. Show that the inverse of the hyperbola with respect to its centre is the 
lemniscate. 

EXAMPLES. 

38. Trace the spiral of Archimedes, p = aS. The distance from the pole is 
proportional to the angle described by the radius vector, tan if> = Q. The curve 
is tangent to the initial line at O. The intercept PQ between two consecutive 
revolutions is constant and equal to 2na. Therefore we need only construct one 
turn directly. The dotted line shows the curve for negative values of 0, which 

* More generally two polar curves are the inverses of each other, when for the 
same their radii vectores are connected by p } p z = k 2 . k = constant. 



162 



APPLICATIONS TO GEOMETRY. 



[Ch. XV. 



is the same as the heavy line revolved about a perpendicular to the initial line 
through O. (Fig. 55.) 





Fig. 55. 



Fig. 56. 



39. Trace the equiangular spiral p = J> . We can write the equation 

= b log p, 

if we prefer, tan ip = b, or the angle between the radius and tangent is constant. 
p — a for S = o, and p increases as 6 increases. p( = )o for = — 00 . 

The pole O is an asymptotic point. (Fig. 56.) 

40. Trace the hyperbolic or reciprocal spiral 
pB = a. The pole O is an asymptotic point.' 
A line parallel to the initial line at a distance a 
above it is an asymptote. For negative values 
of 0, rotate the curve through it about a normal 
to OA at O. (Fig. 57.) 

41. Trace the lemniscate p 1 — 2a' 2 cos 20. 
Fig. 57. 

42. Trace the conchoid p = a sec ± b, 

or (*»+.7 2 X* - a ) 2 = b2x2 - 

When a < b, there is a loop; when a = b, a cusp; when a >6, there are two 
points of inflexion. (Fig. 58.) 





Fig. 58. 



Fig. 59. 



Art. 



'5-J 



EXAMPLES OK CURVE TRACING. 



163 



43. Trace the cardioid p = *(i -f cos 0). The curve is finite and closed, 
symmetrical with respect to Ox. p = 2a, a, o, for 9 = o, \n, tt, and 
diminishes continually as increases from o to it. Also, tan rp = —"cot 10. As 
9(=)it, ii'(=)it, or the curve is tangent to Ox at the pole, which point is a cusp. 
The rectangular equation is 

x* -f /-' _ ax = + a j/* 2 +y*. (Fig. 59.) 

44. Trace the three-leaved clover p = a cos 30. 

45. Trace the curves : 

(1). p = a sin 20, p=acos20. 

(2). p = a sin 30, p = a sin 48, 

(3). p — a sec 2 |0, p = a sec 0. 

(4). p = a sin 0, p = a sin 3 ^0. 

46. Trace the curve p(0 2 — 1) = ^0'-'. 

47. Trace p — a vers and p = «(i — tan 0). 

48. Trace the evolutes of y = sin x and y = tan x. 

49. The Cycloid. The path described by a point on the circumference of a 
circle which rolls, without sliding, on a fixed straight line is called the cycloid. 



y 


N 


L 


V 








R 




G 






X 





? 


j 


A 


I 


D 


A 





Fig. 60. 

(1). Let the radius of the rolling circle MPL be «, the point Pthe generating 
point, M the point of contact with the fixed straight line Ox which is called the 
base. Take MO equal to the arc MP; then O is the position of the generating 
point when in contact with the base. Let O be the origin and x, y the coordinates 
of P, Z PCM = 0. 

Then we have 

x = OM - NM - a(B - sin 0), y = PN = a(i - cos 0). 

The coordinates are then given in terms of the angle through which the rolling 
circle has turned. OA = 2ita is called the base of one arch of the cycloid. The 
highest point V is called the vertex. Eliminating 0, we have the rectangular equation 

x = a cos -1 — ty'zay — v 2 . (Fig. 60.) 

(2). To find the equations to the 
cycloid when the vertex is the origin, 
the tangent and normal there are the 
axes of x and y, we have directly from 
the figure 
x = <?0 -)- a sin 0, y — a — a cos 0. 

Eliminating for the rectangular 
equation, 

x = a cos- 1 -(- \/zay — y 2 . 

(Fig. 61) 
The cycloid is one of the most important curves. 




i6 4 



APPLICATIONS OF GEOMETRY. 



[Ch. XV 



50. The Trochoids. When a circle rolls on a fixed straight line, any point 
rigidly fixed to the rolling circle traces a curve called a trochoid. The curve is 
called the epitrochoid or hypotrochoid according as the tracing point is outside or 
inside the rolling circle. 

Their equations are determined directly from the figure. 




Fig. 62. 



Let 
Then 



CM = a, 
= ON = 



CP = p, CP' = /, / MCP = B. 

zQ — p sin B, y = PN = a — p sin B, 

for a point P on the hypotrochoid PV. Replacing/ by/', the same equations give 
the epitrochoid. (Fig. 62.) 

51. Epicycloids and hypocycloids. 

The curve traced by any point on a circle which rolls on a fixed circle is called 

an epicycloid or a hypocycloid, according as 
the circle rolls on the outside or on the inside 
of the fixed circle. (Fig. 63.) 

Let O be the center of the fixed circle of 
radius a, and C the center of the rolling circle 
of radius />, and P the tracing point. Then 
with the notations as figured, we have 
arc AM = arc PM, or aQ = bcp. Hence 
x = ON = OL - NL, 

= (a 4. b) cos B — b cos (8 -f 0), 

* + b a. 



V 
















M 




^91 


C 




a^ 






v p/ 


K y 




*^£^-— — ■ 

















A\ 


N 


L 



(a -j- &) cos Q — b cos 



b 



CK — (a + b) sin Q - b sin (6 + 0), 



Fig. 63. 
y = PN = CL 

— (a -f- b) sin B — b sin 

for the coordinates of the epicycloid. For the hypocycloid change the sign of b. 

In this book convexity or concavity of a curve at a point is fixed by the sign of 
the second derivative of the ordinate representing the function. D%y = + or 
DyX = -f- means convexity with respect to O x or O y respectively. This is the 
equivalent of viewing the curve from the end of the ordinate at — 00 , instead of 
from the foot of the ordinate as is sometimes done. 



PART III. 

PRINCIPLES OF THE INTEGRAL CALCULUS. 



CHAPTER XVI. 
ON THE INTEGRAL OF A FUNCTION. 

116. Definition, — The product of a difference of the variable 
x 2 — x 1 into the value of the function f{x) taken anywhere in the 
interval (x^ , x 2 ) is called an element. 

In symbols, if z is either of the numbers x x or x 2 , or any assigned 
number between x x and x 2 , the product 

2 - x i)A z ) 
is the element corresponding to the interval (x lt x 2 ). 



Geometrical Illustration. 

If y = f(x) is represented by the curve AB in any interval (a, b), 
and x 19 x 2 are any two values of x in 
[a, b), then the element corresponding 
to (x x , x 2 ) is represented by the area 
of any rectangle x 1 M l Mx 2 , whose 
base is the interval x 2 — x\ , and alti- 
tude is the ordinate zZ to any point 
on the curve segment P x P r 

117. Definition. — The integral of 
a function f(x) corresponding to an 
assigned interval (a, b) of the variable 
is defined as follows: 

Divide (a, b) into n partial or sub-intervals (a, xj, (x x , x 2 ), 
. . • , (^«_ 2 , Xn-i), { X n-i > <^)> by interpolating between a and b the 




numbers 



x H _ l taken in order from a to b. And for con- 



tinuity of expression let x = a, x n = b. 

The integral of a function is the limit of the sum of the elements 
corresponding to the n sub-intervals, when the number of these sub- 
intervals is increased indefinitely and at the same time each sub- 



interval converges to zero. 



165 



i66 



PRINCIPLES OF THE INTEGRAL CALCULUS. [Cn. XVI. 



In symbols, we have for the integral oi/(x) corresponding to the 
interval (a, b), 



x r { = ).v r -l r = n 



In which z r is either x r , x r _ 1 or some number between x r and x r _ lt 
or as we say, briefly, some number o/*the interval (x r _ T , x r ). At 
the same time that n = co we must have x r — x r _ x (-=)o. 

Geometrical Illustration. 

If y = f{x ) is represented by a continuous and one-valued ordinate to a curve, 
then the integral of f(x) for the interval («, b) is represented by the area of the 
surface bounded by the curve, the x-axis, and the ordinates at a and b. 




ind 



For, any elementary area, such as (x 3 — x 2 )/[z s ), lies between the areas of the 
rectangles x 2 P 2 M s x 3 and x 2 N' s P s x a constructed on the subinterval (x 2 , x^), or is 
equal to one of them, according as z z = x 2 , z s =^ 3 . Also, the corresponding area 
i ,P 2 P s x s bounded by the curve P 2 P 3 , Ox, and the ordinates at x 2 and x 3 lies 
between the areas of the same rectangles, in virtue of the continuity of f(x), when 
x :i — x 2 is made sufficiently small. 

Hence the sum of the integral elements and the fixed area of the curve lie 
between the sum of the rectangular areas 

Afo + N 2 x 2 + . . . + Nnb (i) 

M x a + M. r v x + . . . + M n x n _ v (2) 

1 .' ! RQ be not greater than the greatest of the subintervals into which (<7, />) is 
■ In ided. The difference between the areas (1) and (2) is not greater than the area 
of the rectangle BDQR, whose base is RQ and whose altitude BR is equal to the 
difference f{b) —/la) and to the sum of the altitudes of A", J/, , N 2 M 2 , .... N H M H . 
This rectangle BQ has the limit o, since each subinterval has the limit o; and so 
also lias RQ) while its altitude is finite and constant, or does not change with ;/. 

Consi quently the areas (1) and (2) converge to the constant area of the curve 
which lies between them, and so also must the area represented by the sum of the 
cli me nts oi the integral. 

Hence the integral oif(x) for (a, b) is equal to the ana of the curve, as enun- 
i iated. 



Art. nS.] ON THE [NTEGRAL OF A FUNCTION. 167 

118. Evaluation of the Integral of a Function.* — In order thai 
a function shall admit of the limit which we call the integral for a 

given interval, the function must, in general, be finite and continuous 
throughout the interval. 

Should the function be finite and continuous everywhere in the 
interval (a } b) except at certain isolated values of the variable, at 
which singular points it is discontinuous, either infinite or indeter- 
minately finite, then special investigation is necessary for such singular 
values, and we omit the consideration of them. 

We shall assume that the functions considered are uniform, 
finite, and continuous throughout the interval, unless specially 
mentioned otherwise. 

The process of evaluating the limit defined as the integral, in 
§ 117, is called inlegratio?i. 

In evaluating the limit 

£ 2(x r - *,_,)/(*,), x r - ^_ 1 (=)o. 

we are said to integrate the function /"from a = x to b = x n . The 
numbers a and b are called the boundaries or limits of the integration 
or integral. The lesser of the numbers a and b is called the inferior, 
the greater the superior, limit of the integration, f 
In the differentiation of the elementary functions 

x a , a x , log x, sin x, 

and like functions of them and their finite algebraic combinations, 
we have seen that the derivative could always be evaluated in terms 
of these same functions. Not so, however, is the case in evaluating 
the integrals of these functions. The integral cannot be always 
expressed in terms of these same functions, and when this is the case, 
the integral itself is a new function in analysis which takes us beyond 
the range of the elementary functions such as we have defined them 
to be. 

We shall be interested, in this book, directly with only those 
functions whose integrals can be evaluated in terms of the elementary 
functions. 

It can be stated in the beginning that there is no regular and 
systematic law known by which the integral of a given function can 
be determined as a function of its limits in general. 

The process of integration is therefore a tentative one, dependent 
on special artifices. 



* For Riemann's Theorem : A one-valued arid continuous function in a given 
interval is always integrable in that interval; see Appendix. Note 9. 

t Tli e word limit as here employed does not in any sense have the technical 
meaning limit of a variable as heretofore defined. It is an unfortunate use of th'- 
word, retained out of respect for ancient custom. It is contrary to the spirit of 
mathematical language to use the same word with different meanings, or in fact to 
use two words which have the same meaning. 



1 68 



PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVI. 



The systematizing of the artifices of integration is the object of 
this part of the text. 

119. Primitive and Derivative. — If we have two functions F(x) 
a.nd/(x), so related that/(.v) is the derivative of F(x), then F(x) is 
called a primitive of f\x). The indefinite article is used and F(x) is 
called a primitive oi/\x), because if 

DF{x) =f(x), 

then also we have 

D[F(x) + C] = /(.v), 

where C is any assigned constant. 
Any one of the functions 

f\x) + C, 
obtained by assigning the constant C, is a primitive of J\x). The 
primitive of f(x) is the family of functions containing the arbitrary 
parameter C. 

Geometrical Illustration. 
The two curves 

y=iF{x) + C 1% (i) 

y _ j? {x) + C 2 . (2) 

are so related that at any point x their tangents at P x and P n are parallel, and each 
curve has for the same abscissa the same slope. Their ordinate* differ by a c< >n 




Fig. 66. 

stant. Each curve represents a primitive of f(x). Any particular primitive is 
determined when we know or assign any point through which the curve must pass. 

120. A General Theorem on Integration. — If a primitive of a 
given function can be found, then the integral of the given function 
from a to X can always be evaluated. The given function being 
continuous in (a, X). 

Let_/"(.v) be a continuous function in (a, X), and let F(x) be a 
primitive <>f/(.\ ). 

Let x Q = a, x M = X. Interpolate the numbers x x , .... .v„_, 
between a and X in the interval (<?, X), in order from a to X, sub- 
dividing the interval (a, X) into the n subintervals 

(.v,, x x ), (x lt x.X . . • , (x H _ 2 , x H _^, (x H _ lt x H ). 



Art. 120.] ON THE INTEGRAL OF A FUNCTION. 169 

We have the sum 

n 

2 (x r - av_ : ) = X - a t 

r = i 

whatever be n. 

Since /(.v) is the derivative of F{x), 

F'{x) =/(.v). 

By the law of the mean value applied to each of the subintervals, 
we have the n equations 

F{X) - F(x„_,) = (,v„ - x„_,)/{g„), 
F(x n _,) - F(x„_ 2 ) = (x„_, - x„_,)/{ £„_,), 

F(x t ) - F{x x ) = (x,_ - x,)AS,), 

F( Xl ) - F{a) = (,v, - ,v )/( gl ). 
Adding, we have 

F(X) - F(a) = 2 (.vv - x r _,)f($ r ), (1) 

r = 1 

in which B, r is some particular number in the interval (x r , x r _ x ). 

The sum on the right, in the above equation, is equal to the 
member on the left. The left side of the equation is independent 
of n. The equation is true whatever be the integer n, and when 
71 = 00 . The sum on the right remains constant as we increase n, 
and being finite when n = co , 

F{X) - F(a) = £ "£(x r - x r _,)/(a r ). 

« = cc r=i 

Now let z r be any number whatever of the subinterval (x r , x r _ x ), 
for each subinterval. Then 

ASr) = A*r) + e r , 

where e ( = )o, when x r — x r _ I (=z)o, by reason of the continuitv of 

Ax). 

Therefore 

n n 

— 2(x r - x r _ x )/{z r ) + 2(x r - x r _^e r . 

1 1 

Let e be the greatest, in absolute value, of the numbers 
e l9 . . . , e n . Then 

n n 

2{x r - x r _ 1 )e r \ -£. I e 2(x r - x r _ x ) = e(X - a), 

1 1 

the limit of which is o, when n = -*> ; provided each subinterval 

x r - x r _ x (=)o 
when n = 00 . 



17° PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVI. 

Therefore, when n r= oo , and at the same time each subinterval 
x r — x r _ l [ = )o, we have 

£ 2(*„ - x r .,V(S r ) = £ 2 (x, - -v r _,y( 2r ), (2) 

w = oo r = i n - x r=i 

2 r being any number of the interval (.r r , ^ r _ x ) ; that is, z r may be 
x r , or ^ r _ t , or any number we choose to assign between x r and 

The member on the right in (2) is, by definition, the integral of 
/(x) from a to A", and we therefore have for that integral 

£ 2(x r - x^,V(z r ) = F(X) - F(a), 

m = oo r = 1 

which is evaluated whenever we know a primitive of f[x), and can 
calculate its values at a and X. 

Observe that it is not necessary that we should know the values 
of the primitive anywhere except at the limits a and X. The integral 
is therefore a function of its limits. 

121. In the preceding articles of this chapter we have fixed no 
law by which the values x v . . . , x n _ J were interpolated between a 
and X. The integral has been denned and evaluated for any distri- 
bution of these numbers whatever, subject to the sole condition that 
the intervals between the consecutive numbers must converge to o 
at the same time that the number of the subintervals becomes 
indefinitely great. 

Since it makes no difference how we subdivide the interval of 
integration, we shall generally in the future subdivide the interval of 
integration into n equal parts, so that 

x r — x r _ l = Ax r = h = (X — a)/n, 

and we shall take the value of the function to be integrated at x r _ t , 
the lower end of each subinterval. 

The integral olf(x) from a to X is then 

F(X) - F(a) = £*2f(x,.)Jx. 

But observe that 

f{x r )Ax= F'(x r )Jx = dF(x r ). 

Hence the integral of /[x) from x = a to x = X is the limit of the 
sum of the differentials of the primitive function. 

122. Leibnitz's Notation. — The notation previously used to 
represent the integral, while valuable as indicative of the operation 
ad initio performed in evaluating this limit, is cumbersome, and when 
once clearly assimilated it can be replaced by a more convenient and 
abbreviated symbolism. We replace the limit-sum symbol by a 



Art. 123.] ON THE INTEGRAL OF A FUNCTION. 171 

more compact and serviceable symbol designed by Leibnitz. Thus, 
in future we shall write in the suggestive symbolism 

(A*)** m £ Zf{x r )Jx, 

^ j w = » r = 

as the symbol for the integral oif(x) from a to X. 

The characteristic symbol / is a modification of the letter 6", the 

initial of sum, and is taken to mean limit-sum, or / = f2. The 

symbol f(x) dx represents the type of the elements whose sum is 
taken. 

If F{x) is a primitive oifyx), then 

F{X) - F(a) = JAx) dx, 
— C F'{x) dx, 

= fjF(x). 

This, then, is the final reduction of the integral; and whenever the 
expression to be integrated, f(x) dx, can be reduced to the differen- 
tial dF(x), then F(x) is recognized as a primitive of f{x) and the 
integral can be evaluated when the limits are known. 

123. Observations on the Integral.— Differentiation was founded 
on the exceptional case in the theorems in limits, wherein we sought 
the limit of the quotient of two variables when each converged to o. 

We found that the theorem stating: the limit of the quotient is 
equal to the quotient of the limits, did not hold, § 15, V (foot- 
note) in the case when the limit of the numerator and of the denomi- 
nator was o, but that the limit sought or defined was the limit of 
the quotient of the variables. 

Integration is founded on another exceptional case in the theorems 
in limits. Here we seek the limit of the sum of a number of terms 
when the number of terms increases indefinitely and also each term 
diminishes indefinitely. The limit we seek is the li?nit of the sum. 
The theorem which states: the limit of the sum of a number of 
variables is equal to the sum of their limits, was only enunciated and 
proved for a finite number of variables, and does not necessarily hold 
when that number is infinite. The sum of the limits of an infinite 
number of variables, each having the limit o, is o and nothing else. 

The important point in the definition of the integral which makes 
it a matter of indifference where in the subinterval of the integral 
element we take the value of the function, is an example of an 
important general theorem in summation, which can be stated thus: 



172 



PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVL 



Lemma. If the sum of ?i variables u x , . . . , u n has a determinate 
limit A when each converges to o for n — oo , so that 

£( Uj + ...+«„) = A, 
and there be any other n variables v lf . . . , z>„, such that each con- 



verges to o for ;/ — oo , and at the same time 

= I, 



then also 

£(V X + . . . + Vn ) = ^ 

For, whatever be r, 

*- = « + e„ 

u r 

where e r ( = )o, when ?i = oo . Also, 

;£^r = ;£^(«r + e r u r ) = £2u r + £2e r u r . 
If e is the greatest absolute value of e 1? . . . , e M , then 
^e r # r | ^ | e2u r — eA, 
the limit of which is o, and, § 15, III, 

£Sv r = £2u r = A. 
This principle is of far-reaching importance in integration, and 
will be frequently illustrated and applied in the applications of the 
Calculus. 

Geometrical Illustration. 

Let y = F(x) be represented by a curve, and let F'(x) = f(x). Then _f(x) is 
the slope of the curve or of its tangent at x. 

We have SQ equal to 

s F(X) - F { a) = M V P X + M. 2 P 4. 

yM n B, (i) 

= 3 JF. 

Also, the sum of the differentials of F 
at a, x-y, . . . , is 
Q 2dF= M x T x -f M 2 T 2 + . . . +M n T n . (2) 

The difference between this sum and 
that in (1) is 

2dF- 24F=r x T l -{-P 1 T t + . . . +£T n . 

But we know that the limit of 

AF _ M r Pr 

dF ~ M r T r 

i- 1 when n = ao and Ax( = )o. Hence, by the lemma above, we have 
F(X) - F(a) =£2JF = £2 dF, 
= £2Ft(x)dx, 

wln'i li is another illustration of the integral. 




Art. 124.] ON THE INTEGRAL OF A FUNCTION. 173 

124. The Indefinite Integral. — When we know a primitive of a 
given function we can integrate that function for given limits. It is 
therefore customary to call a primitive of a given function the 
indefinite integral of that function. 

Indefinite integration is therefore a process by which we find a 
primitive of a given function. A primitive F(x) of a given function 
f{x) is called the indefinite integral of f(x), and we write conven- 
tionally, omitting the limits, 

f/(x) dx = F(x). 

This, of course, becomes the definite integral 

j/( X )dx = F(X) -F(a) 

when the limits of integration a and X are assigned. 
The indefinite symbol 



fA*) 



dx 



proposes the question : Find a function which differentiated results 
mf(x); or, find a primitive oif(x). 

Before we can solve questions in the applications of the integral 
calculus, we must be able, when possible, to find the primitive of a 
proposed function. The next few chapters will be devoted to this 
object. 

125. The Fundamental Integrals. — The two integrals 

x x 

(' e x dx and j sin x dx 

J a v a 

are called the fundamental integrals. They can be determined 
directly by the ab initio process, and all other functions that can be 
integrated in terms of the elementary functions can be reduced to 
the standard form 



/ du = u 



by means of these fundamental integrals. 
1. We have, where (X — a)/n = h, 



I 



x 

e*dx = £ h[e a + e a+h +..... + *»+«•-«>*], 

h(=)o 

p*h T h 



v 



174 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVI. 

2. Also, 
x 

Jf sin xdx = ^"/^[sin a -J- sin (6? -f- ^) -|- . . . -|- sin (a-\-n—ih)], 
a A( = )o 

sin [a -(- J(« — i)h] sin \nh 
~^ "sin ±h ~' 

by a well-known trigonometrical summation.* 

But the expression under the limit sign is equal to 

{cos {a — ±h) — cos [a -f- £( 2 » — i)A] j 



sin \h 



= (cos (a - iA) - cos (A" - i*)\J^, 

which, when ^( = )o, has the limit cos a — cos X. 
x 
C sin x dx = — cos X -\- cos a. 

* See Loney's Trigonometry, Part I, § 241, p. 283. 



CHAPTER XVII. 
THE STANDARD INTEGRALS. METHODS OF INTEGRATION. 

126. As stated in the preceding chapter: ii/{x) is the derivative 
of F(x) i then F(x) is a primitive of f{x) y or an indefinite integral 
of f{x). This and the next chapter will be devoted to finding primi- 
tives of given functions.* This process is nothing more than the 
inverse operation of differentiation. The word integrate, when used 
unqualified, for the present means " find a primitive." 

If we choose to work in derivatives, then in the same sense that 
Df{x) means, find the derivative oi/(x)\ the symbol D~ l /[x) means, 
find a primitive oif(x). 

It is usually preferable to work with differentials and employ the 

symbol lf(x) dx to mean, find a primitive of f{x), or simply, 

integrate /%*:). 

If u is any function of x, then 



u = I du 



and is the solution of the integral, 



The solution of 



f/{x) dx 



invariably consists in transforming f(x) dx into the differential du 
of some function u of x, and when this is done the integral or primi- 
tive u is recognized. 

But, inasmuch as every function that has been differentiated in 
the differential calculus furnishes a formula, which when inverted by 
integration gives the corresponding integral of a function, we do not 
consider it necessary that we should always reduce an integral com- 
pletely to the irreducible form / du. There are certain standard 

functions, such as those in the Derivative Catechism, which we select 
as the standard forms whose integrals we can recognize at once, and 
thus save the unnecessary labor of further and ultimate reduction to 

du. 



I 



* This is the starting-point of the theory of differential equations, an extensive 
branch of the Calculus. 



75 



176 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 

The Integral Catechism. 

1. / cu dx — c I u dx. 

2. f (u + v)dx = fu dx + fv dx. 

3. / it dv = uv — / v da. 



A f J Ua + l 

H-. / u a du = . 

J a + I 

5./^ = log*. 

6. I e u du == *•". 

7. / a" du = — — . 

J log a 

o C ■ j cos au f sin 

8. I sin #« du — — . I cos <z« #« = 

J a J a 



a j* = i. 



cot aw 



. / sec 2 au du = . / esc 2 au du = — 

10. / sec u tan u du = sec w. / esc 7/ cot « ^ = — esc «. 

.. p du u u 

11. / — ■ = sin -1 — = — cos— 1 — . 
J |/ a 2_ M 2 a a 

12. (—£= = log (« + V^T^ 2 ). 
^ y « 2 ± «' 2 

^^ r du i « 

13. / -^-r— 2 = - tan-i-, 
»/ zr -L- a 2 a a 

.. /* du i , u — a I , a — u f du 

•4. / -5 - n = — log — — , or — log — ■ — = / -. 

J u* — a 2 2a & u -f a 2a * a + u J a 1 — u l 

15. I tan u du = log sec u. I cot u du = log sin u. 

16. / sec u du — log tan (\u -\- %it). I esc u du = log tan \u. 



du i « i « 

or cot— i — . 

a ' a 



17. f \/a 2 - u 2 du = \u \/a 2 - u 2 -f ±a 2 sin-' -. 

18. / ^u 2 ± a 2 du — \u \/u 2 ± a 2 ± \a 2 log (u -f \/u 2 ± a 2 ). 

19. / sin 2 u du = \u — ± sin 2u. I cos 2 u du = \u -f- ^ sin 2tt. 

20. / log » </* = u{\og u — I). 

p du I « I 

21. / := = — sec- 1 - = esc 

J u a/ u i _ a 2 a 



du - I U I M 

== = — sec -1 - = esc -1 — 

|/ w 2 _ a i a a a a 

du 

4/2;/ 



22. / - = vers- 1 u = — covers -1 u, 



Art. 127.] METHODS OF INTEGRATION. 177 

These standard forms are certairTelementary junctions of frequent 
occurrence, and they constitute the Integral Catechism, which should 
be memorized, and to which must be reduced all other functions 
proposed for integration. . 

In the formulae, u. i\ etc., are functions of x. 

. 127. Principles of Integration. — The first two formulas in the 
Catechism enunciate two fundamental principles of integration. 
I. Since c du = d(cu), where c is any constant, we have 

/ c du = I d(cu) = cu = c I du, 

or the integral of the product of a constant and a variable is equal to 
the product of the constant into the integral of the variable. There- 
fore a constant factor may be transposed from one side of / to the 
other without changing the integral. 

EXAMPLES. 

1. fx 3 dx=*fx 3 dx = l f(±x*)dx = i / d(x±) = ±x*. 

« C ■ j * f, ■ \j l C j, x cos ax 

2. I sin ax dx = / (— a sin ax)dx = / d(cos ax) = . 

II. Since d(u -j- v -j- w) = du -f- dv -f dw, 

j {du -\- dv -j- dw) = I d(u -f- v -j- w), 



= u -4- v 



= J du-\- dv -\- I dw. 

It follows, therefore, that the integral of the sum of a finite number 
of functions is equal to the sum of the integrals of the functions, and 
conversely. 

EXAMPLES. 

1. / (ax -f cx*)dx — I axdx-\- I ex* dx, 

= a I x dx -j- c I x z dx, 

= ajd(lx*) + cfd(lx*), 
= lax 1 + Jcc*. 

2. / (cos x — sin ax) dx — I cos x dx — I sin ax dx, 

-/*(-■> + /<<=?). 

= sin x -\ cos ax. 

a 



1/8 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 

128. Methods of Integration. — The first and simplest method of 
integrating a given function is, when possible, to 

Complete the Differential. 

This means, to transform the integral into / du by inspection, and 

thus recognize u. Except for the simplest functions this cannot be 
done directly, and we have recourse to the following. 

The methods employed by which we reduce a proposed function 

to be integrated to the irreducible fundamental form / du, or to the 

recognized form of one of the standard tabulated functions in the 
Catechism, are 

I. Substitution. 

(1) Transformation. (2) Pationa/ization. 

II. Decomposition. 

(3) Parts. (4) Partial Fractions. 

129. While nearly all the standard integrals in the catechism are 
immediately obvious by the inversion of corresponding familiar 
formulae in the derivative catechism, we shall deduce them by aid of 
the principles of § 127 and the methods of § 128, and the two 
fundamental integrals 



/ e* dx = e x , / sin x ~ 



cos x, 



established in § 125, in order to illustrate the methods of integration 
laid down in § 128, and to fix the standard integrals in the memory. 

130. Transformation [Substitution). — This is a method by which 
we transform the proposed integral into a new one by the substitu- 
tion of a new variable for the old one. The object in view being to 
so choose the new variable that the new integral shall be of simpler 
form than the old one. 

Thus, if the proposed integral is 



j/(x) dx, 



and we put x = cf)(z), then dx = <ft'(z) dz. The integral is trans- 
formed after substitution into the new integral 



//!>(*)] </•'(*) dz. 



This when integrated appears as a function of z, which is retrans- 
formed into a function of x by solving x = cp(z) for z and substituting 
this value z = ip(x). The final result is the proposed integral 



y>(.v) dx. 



Art. 130.] METHODS OF INTEGRATION. 179 



1. Use a substitution to find 



EXAMPLES. 

du 



/du 
u ' 



Put u = f v , then du = e° dv, 

>du 



2. Make use of / e 11 du — e u to find / 1 



U a du. 



Put u 1 = £•"'. .*. au a ~ l du = e°dv. Hence 

a— 1 



«<* <&< = — e v u dv — — e a dv 



a a a 

e a u a + 1 

a-\- I a-\- 



^ a ,/ ( ! j, \ 

M \ « / 



/ « a </« = 



3. Integrate / cos x dx, given / sin u du = — cos u. 

We have cos .*■ dx = — sin (-J-7T — x)d(\it — x). 
Hence, if u = i-7t — x, 

I cos x dx = — / sin « ^ = cos u = sin x. 

4. Integrate tan x dx. We have, by Ex. 1, 

//•sin x , /V(cos jr) 
tan x dx — / dx = - = - log cos x. 
J cos x J cos x 

5. Integrate / cot x dx. 

/» , /V(sin x) . 

cotxdx= I —. ' = log sin x. 

J J sin x 

6. Show that 

/ sin ax dx = cos ax ; / cos ax dx = — sin ax. 

7. Show that / tan ax dx = - log sec ax. 

dx 

tfl — x 

Substitute x = sin z. .-. dx = cos z dz. 

— X = dz = z = sin- 1 z. 
\/i - x* J 

9. Integrate / (a + £*)' </.*. 

Put -f- &r == y. .-. dx — dy/b. 

10. f ^ „ . Put / = fltan0. Then 
./ <z 2 -|- / 2 

<# = a sec 2 6 <#. Hence 



/ax 
— , 
4/ 1 - 



/ -= , = — I dB = — = — tan-* — 

J a* -]- t 2 a J a a a 



8o PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 

11. / dM du. Put/ = a u . .'. dy = a u log a du. 

/ A« du = / dy = ■ = ■ . 

J log « «,/ iog « log a 

he functions 

*, x 2 - 2* a + ^ + r 2 *. $ ^ _ e~T) . 



12. Integrate the functions 



13. Integate 



X -f I ' X 3 -f I ' JT" -|- «" ' 

fcosxdx /V/(i + sin *) . 

14. / : : = / -*— - . - = log (I -4- SID *). 

«/ I -f- sm x ,/ I -f sin x ° v ' 

15. / sin 2 xdx — £ I (I — cos 2.v//.r = \x — |sin 2jc. 

/sin x dx /' 
. / cos 2 x dx. 
I — cos X J 

17. Given the definite integral 

/ /^// — = _ 

Ja C + T Ja '+ 1 

fdt 
feduce the integral f — = log t. 



In the value of the definite integral, let c{ — ) — I, then (see § 75, 0/0 form), 
1 i = £{x^ log x - ac+i log a), 



*( = )- 



= log x — log a. 
log « is the constant of integration and we have 

/*■ = ***• 

~ /' du 

18. / . Put u — a sec 9. 

' u \/u 2 — a 2 

19. / — — This can be written 
J \/2S — J 2 

r ds 

J |/i - (1 - sf 

Put 1 — s = cos 0. . •. *fo = sin d$ } and the integral becomes 

/ dO = = cos— !(i — j) = vers— " s. 

on I 1 , P sin x /» </(cos x) 

20. / sec .r tan x dx = J dx — — \ — , 

J ,' cos 2 X J COS 2 X 

= sec x. 

21. / cs t a d\ = ? 

22. f- =. Put |/jc 2 -f a 



j — jr. .-. dx = dz. 



■• I —7^ ; = / -^ = log z = log (x -f |A- 2 -f a 2 



Art. 130.] METHODS OV INTEGRATION. 1S1 

23. Show by a like substitution that 

« 7 \ x- — a 2 

24. Integrate / . 

,' sin a cosjt 

f d* Psec*xdx f d (tan x) . , 

J siu-T-coTTv =J lainr =J -te^r = g (tan x) ' 

25. / = / . . v ■ = log (tan i.v), by Ex. 24. 

«/ sin jr J sin |.r cos 4-x s v - ; y * 

26. To integrate / , put x = \n — z 

J COS X 

/dx n dz 

^rx = -j-^r z = - lo * < tan w = iog(cat ^), 



= log[cot (£tT - £*)] = log tan (U -f \n). 
These results can be identified with 16 in the table. 
27. Observing that we can, by inspection, write 



_ L_ = 1 (_E L_\ 

x 2 — a 2 2a \x — a x -\- a J 



we have 

dx 



/■_^_ = i log 

J x 2 — a 2 2a . 



2a x -\- a 

This process is a particular case of the general method of decomposition into 
partial fractions. 

Integrate this case, using the substitution {x — a) = (x -j- a)z. 
Also, integrate the more general integral 

f dx 

J (x — a)(x — 6)' 

by means of the transformation x — a = (x — b)z. 

28. We can make use of Ex. 27 to obtain the integrals in Exs. 25, 26. For 
we have 

/dx _ rcosxdx _ /* ^(sin x) I /i -f- sin x \ 

cos x ~J cos 2 jc — e/ I — sin 2 x ~ 2 \i — sin x J ' 

Show in like manner that 

J sin x 2 \i -f- cos x/. 

— . Put ** = sec 9. 

j/* 2 * - 1 

Then dx = tan 6 dQ, and the integral becomes 

fdQ = = cos-* (*-*). 

30. Integrate / . 

J a — b cos 

We can complete the differential by inspection, for the integral becomes 

I pd{a — /;cos 6) 

~b 



rd(a - bcosb) 1 , , . „ 

/ -i j g-' = - log (a - b cos 9). 

J a — b cos Q b 



or 



2 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 

Otherwise, put a - b cos — z. .-. b sin dfj = dz. 
The integral is therefore 

tH = ^ logc= T log {a ~ b cose) * 

31. f — dx . Put x 2 ± a 2 = s 2 . Then x^r:i <&, 
^ \/x 2 ± a 2 

dx _ dz _ ^ -f- ffe _ d{x -f- 2) 

z ~ x - ' z + •* ' X -\- z ' 

i> dx [dx _ Pd(x -f z) _ 

'"' J \tx 2 ± a 2 = J z ~ J x + z~ _ ° g + 

= log (* + |/** ± a 2 )- 

• 131. Rationalization (Substitution). — The object of this process 
is to rationalize an irrational function proposed for integration, by the 
substitution of a new variable. 

Rationalization by substitution is but a particular case of trans- 
formation by substitution. But, since the direct object in view in 
rationalization is not generally to reduce the function directly to a 
standard integral, but to first transform it into a rational function 
which can be subsequently integrated by decomposition into partial 
fractions, the process demands separate and distinct recognition. 

Only a few simple examples will be given here in illustration. 
The subject will be considered more generally in the next chapter. 

EXAMPLES. 

1. Integrate f(a -f bx*fx* dx. 

Put a -f- bx z = z 3 . . \ bx 2 dx = z 2 dz. On substitution the integral becomes 

?/<* -«*>*=£(? -4)' 

(a -f- b^)\ S bx* - 30). 



2. Integrate / 



40b 2 
dx 



(a + bx 2 f 

Put a + bx 2 = z 3 . . \ x dx = 3c 2 dz/2b. 

3 

The integral is — ty a + bx 2 . 

3. Put a -f- bx — c 3 , and show that 
x dx 



f 



{a + 6 x )i \ b 



= ~r 2 (bx - la){a + bx)\ 



4. Put a 2 — x 2 = c 3 , and show that 



/ 



_ l( 3 a 2 -\-2x 2 )(a 2 - xrf' 



(a 2 - x 2 f 2 ° 



5. To integrate / 



dx 



x* \/i -f x 3 



ART. 132. J METHODS OF [NTEGRATION. 183 

Put 1 -f 1/.1-2 = z\ .-. dx = - **» dz. 
The integral becomes 



/I 2X 1 — I 

6- / , . • Put l/x* - I = z*. .: dx = - x* z dz. 

»' X 2 f/I — x' 1 

After substitution the integral becomes 

7 A 1 + •*"*) <&:. Put ^ = z*. . .-. </* = 423 <&. 

" ' 1 - x± 
The integral becomes 

/*s 3 (i + z) , r z*dz ( z* z 2 

VT^-* = - 4 iz"=-*{l+l+ z + lo 8< 2 -') 



-i- = * + »+, + _! 

2 — I ' 2 — 



•(i -f X 1 )^ 



.-. / ! — = - -x J -2^_4r^_4log (x<- 1). 

J I - x* 3 

/ |/a 2 — x* dx. Put * = a sin 6. . •. dx — a cos </0. 

•'• J tfd* - x 2 dx = a 2 J cos 2 6 dQ = \d> J (1 + cos 20)</0, 



id\Q + ^sin 26), 



= Ira 2 sin-i hi-* r a ' 2 — x 2 - 

Rationalization by trigonometrical substitutions will be considered more gener- 
ally later. 

132. Parts [Decomposition). — This important method of decom- 
posing an integral into two parts, one of which is immediately inte- 
grable by definition and the other is an integral of more simple form 
than the original integral, is one of the most powerful methods of 
integration we possess. It is based on the formula for the differentia- 
tion of the product of two functions, 

d{uv) = u dv -\- v du. 
u dv = d[uv) — v du. 
Integrating, we have the formula for integration by parts, 

/ u dv = uv — I v du. 

EXAMPLES. 
1. Integrate / log x dx. 

Decompose the differential log x dx, so that 

u = log x and dv = dx. 

J dX A 

du — — and v = x. 

x 



1S4 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 



Hence 



/ log x dx — x log x — I dx = x log x — x. 
2. Integrate / tan- 



ijr dx. 



Put 


u = tan- 


l x, dv = dx. 


Then 


*- dx 






i + 


X 2 




/ tan- J jc 


dx = x tan— 1 a 
= x tan - l x 


3. Integrate 


/ x e* dx. 




Put 


it = X, 


dv = e* dx. 


Then 


du = dx, 


v = e x . 



H 



+ ^ 



/ x e x dx = x e x — I e x dx = e*{x — I). 

4. Integrate 1 x a log x dx. 

Put /< — log x, dv = x a . 

, dx j"+i 

. \ du = — , v = . 

x a -+- I 

• •• / xa lo g xdx = — _ log x - f X * , 
log* 



*'+ I ° (a + i) 2 

5. Integrate j \/x 2 -f a 2 dx. 

Put « = |/x 2 + a 2 , dv — dx. 

x dx 



du = 



S/x 1 + d 1 
Hence 



f |/*2 + ^ </* = jr |/x 2 + «2 _ f x2dx m 
•> J x/xi+d* 

But 

f <&++* = f-^Ldx = f *** + (-**?- 

Adding, we have 

2 /* i/x^d 2 dx = x \/x* + « 2 -f a 2 /* — <fa - 
' «/ i/^-j-a 2 ' 

.-. J V^ 2 + a 2 dx = fr ^V 2 + «2 + ^a 2 log(A- + |/I :2 >^), 

by Ex. 22, § 130, or Ex. 31, § 137. 



Art. 133. J METHODS OF INTEGRATION. 185 

6. Show, in like manner, that 

I tfx* — a* dx = U- \/x 2 - d l - \a* log (x + \/x 2 -a*). 

7. We can frequently determine the value of an integral by repeating the process 
of integrating by parts. Thus, integrate 



/• 



e ax sin bx dx. 

Put u — sin bx, dv — e ax dx. 

du = b cos bx dx, v = —e ax . 
a 

/e"* sin bx dx = — e ax sin bx / e ax cos bx dx. 
a a J 

But, in the same way, we have 

/ e ax cos bx dx — — e ax cos bx + - / e ax sin bx dx. 
Substituting and solving, we get the integrals 



e ax sin bx dx — — (a sin bx — b cos bx), 

e ax 

e ax cos bx dx — (a cos bx 4- b sin &r). 

a 1 -\- b' 1 v 



Put £/« = tan a, then these integrals can be written 

sin (for — a) and — === cos {bx — a) 



\/d 2 -f b 2 \Za 2 + ^ 

respectively. 

8. Use Exs. 5, 6 to integrate 

/x 2 dx p x 2 dx 
— - and / . 

tfx* _|_ a 2 J \/x 2 - a 2 

9. Show that / sin—** dx = x sin- 1 .* -\- \/i — x 2 by putting u = sin- 1 *, 
dv = dx. 

10. Use the method of Ex. 5 to show that 

/ \/a 2 — x 2 dx = \x\/d 2 — x 2 -f- \a 2 sin-*—. 

11. Use the work of Ex. 10 to get 



/ 



x 2 dx 



4/« 2 



— — \x \/a 2 - x 2 -f- la 2 sin-i— . 



133. Rational Fractions (Decomposition). — Whenever the func- 
tion to be integrated is a rational algebraic function, we know from 
algebra (see C. Smith's Algebra, § 297) that it can always be decom- 
posed into the sum of a number of partial fractions, each of which is 
simpler than the proposed function. (See Chapter XVIII.) 

We do not propose to consider here the general process of inte- 
grating rational fractions, but merely consider a few elementary 
examples illustrating the process. 

If the function to be integrated is the rational fraction 

(t>(x) 



1 86 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 

and the degree of is higher than that of ip, we can always divide 
by ip, so as to get 

in which the quotient f(x) is a polynomial inland can be integrated 
immediately. The remainder F(x)/tp{x) is a rational function in which 
F(x) is a polynomial of one lower degree than tp(x), the general 
integration of which will be considered later. 

EXAMPLES. 

— — X 3 X 2 -(- x — log ( I -f- X). 

3 2 

e/ * 2 + 4 



^ l0g ^f-2-2- l0g( " 2 - 4) ' 



dx. 



3* + *- x _?? r lJ ==x + i 



* 2 + 4 * 2 + 4 -*" 2 + 4 * 2 + 4 

* * J * 2 + 4 ax -J xax +j3* + 4 2 J * 2 + 4 ' 

= — .r 2 + — tan-i — — — log (x l + 4). 

2 2 2 2 & V ' ^' 

4. To integrate / T . 

& J (x — a)(x — b) 

We can always write 

« = -1-1-1 L_\ 

(x — a)(x — b) a — b\x — a x — b) 
by inspection. Therefore 

/dx I . x — a 
= log . 
(x — a)(x — b) a — b x — b 

134. Observations on Integration. — The processes of Substitution 
and Decomposition, in their four subdivisions: 

1 . Substitution, 

2. Rationalization, 

3. Parts, 

4 . Partial Fractions, 

constitute the methods of finding a primitive of a given function by 
reduction to a recognized or tabular form. These may be regarded 



Art. 134.] METHODS OF INTEGRATION. [87 

as the rules of integration in general form corresponding to the rules 
of differentiation. With this difference, however, that in integra- 
tion there are no regular methods of applying these rules to all 
functions as is the case in differentiation. 

The successful treatment of a given function depends on practice 
and familiarity with the processes of the operation. 

Sometimes different processes of reduction lead to apparently 
different results. It must be remembered, in this connection, that 
the indefinite integral found is but a primitive of the function pro- 
posed, and both results may be correct. They must, however, differ 
only by a constant. 

Frequently, in reducing an integral to a standard form, we shall 
have to use all four of the methods of reduction. Experience soon 
teaches the best methods of attack. 

In the next chapter we shall consider the subject more generally 
and make more systematic the methods of reduction to the standard 
forms. 

EXERCISES. 

Integrate Exs. 1 to 10 by the primary method of completiag the differential 
by inspection. 

1. I x* dx, / ax— 3 dx, i 2x~*dx. 

2. f(x 2 + I)* xdx = \{x* -f 1 f. 

m x i _ a r )dx x 

4. f(iot* - t-*)dt = 6t* + ±t~ z . 

5. f(x~i -f x~i)dx, f(s 2 - i)ds/s, fv dv/(v 2 — 1). 

6. / r , * du — lo g V u * + 2U - 

J tl 2 -j- 2U 

7. A/2 _ 2ft-* dt = 2;-* - 6/- 2 + \t 2 - log / 6 . 
C{a 2 - x 2 f \/xdx, f{Va- ijxfdx, f (x + ifdx. 



8. 



/» 2ax + b j f e* — <^ x j f 2ax -\- b 

9 " J ax 2 + bx + c ' J e* + e~* ' J {ax 2 + bx + cf *' 

//» e x p seclr , 

J tan-Jjr J sin- 1 * J log x x 

10. Write immediately the integrals of 

I x x x 1 x n ~ l 



x+i' x + l' x 2 + 1 ' *? + 1 * w + «* 
cos 2 -^jr, cos - ^ sin .r, tan M ^r sec 2 *. 



:SS 



*•/ 



cos mx cos nx dx = 



sin [m -\- n)x sin (m — n)x 



Put 


x -- 


— ~2 # 


Put 


e x 


= Z. 


Put 


x n 


= Z. 


Put ] 


Og X 


= z. 


Put 


X 2 


■=. z. 


Put 


X* 


= z. 



PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 

, /' cos \^X . . ,— 

1. / = — dx — 2 sin y x . 

J \?x 

2. / e* cos e x dx = ? 

3. / nx"- 1 cos x M dx = 7 

.. f 2X 

5 -./r+^ = ? 

6. /"_3^_dS r = ? 

e/ I + JC b 

- f dx f dx Pu dv -\- v dn 

J 4/1 — 4jf 2 ' ./ |/L— 2x 2 ' «J |/i — #V 

' J I + 4^ 2 ' «/ 9^' + 4' ./ ^r-j/fe 2 — 1' 
9. / sin 3.r </x, / sec 2 40 </9, / cos \<pd(£>. 

— - — =— = — log (a 4- &*«). 

a -j- bx» nb 5 v ~ y 

J a + bx*~ |/^ an \ \|a/■ 
/'(I4-x 2 ) 2 </.A: 

22. J ^^^ = ^g x + x 2 + J**. 

23. / l jL— =2tfx+* 

'' x\/x \/x 

24. j tan 2 </0 = tan (p — (p. /cot 2 $ d(f> = ? 

25. / sin 20 </tf = ? /"cos 2QdS = ? 



2(m -)- «) 2(w — «) 

C ■ . , sin (in — n)x sin (w + n)x 

I sin ?/u sin «;r our — - : : : ! i_. 

J 2 {m — n) 2(111 4- n) 

Use cos a cos fi = $ cos (a -j- /j, -f- £ cos (a — |8), etc. 

nn C • j cos (W -(- ll)x . COS (W — ll)x 

2*. — I Sin wjt cos «.* </.r = '— 4- ■ - — . 

»/ 2(111 -j- ;/) 2(w — n) 

28. I sin |x cos \xdx = ? / cos 3^ cos 5a: <&.- = ? 



29. 



flog .r 



• J -J^ ^ = M lo g *?• 



14 



^' / -\ ~ *** — * sin_i — V a 



Art. 134] METHODS OF INTEGRATION. [89 



Multiply the numerator and denominator by 4/ 'a -j- x. 

31. / xtfx -j- a dx - \{x + a)S - \a{x -f a)*. Put # + a = z\ 

32. \x*e*dx = e*(x 2 - 2* + 2). Parts. 

33. / *V* aJr = e*{x* - 3^ -f 6* - 6). Parts. 

_. C dx 2 \2x — a „ 

34. / _____ = - tan-i . Put 2ax — a 2 = z 2 . 

J x \/2ax — a 2a \ a 

35. / cot-** dx = x cot-'jr -f 4- log (I -f- x 2 ). 

36. / x tan-'x dx — . ±(x 2 + 1) tan~i_ — \x. 

37. / x 2 sin x dx — 2 cos x -\- 2x sin x — x 2 cos x. 

38. / x 2 cos .r dx = x 2 sin .*• -j- 2.* cos x — 2 sin .r. 

39. / cos .r log sin x dx = sin .* (log sin .r — 1). 

40. / xe ax dx = e ax — . 

J a 1 

.* f dx r dx P dx 

' J (x - i)(x + 3)' J x 2 + sx - 10' J 3* 2 -3-^-6* 

Hint. Complete the square, 
^r 

j/(_ - a)(_ - /J) 
Put x — a — s 2 , then dx = 2z dz 

dx C dz 



= 2 log (\/x - a + Vx - (3). 

a/i.t — rtMx — fi\ 



.-. f dX = 2 f dZ 

J tf{x - a)(x -ft) J \/z 2 -f- a - ff 

= 2 log (z 4. ^ + a- /?). 
43. A ^ = 2 sin- |_E_. 

Put x — a = z 2 , as above, and the integral becomes 

2 f dZ . 

^ V/5 - a - 2 2 

44. r i/a + 2^ + c^ 2 ^ 

a -j- 2<ir -j- ^* 2 = c—T-(ac -4- 2bcx -f- r 2 x 2 ), 

= r-i[rrjr -f ^) 2 — (b 2 — ac)\ 
Put ex -f- (5 = 2. . •. flfjc — r/z/V, and the integral becomes 

-^ /* |/^ 2 -{b 2 - ac)dz, 
the standard form 18, § 126. 



190 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 

/</.v 
; — , where m and n are positive integers, 
(* — ay»(x - />)" l 5 

or m 4- >l is a positive integer greater than i. 

Put x — a = (x — b)z, then 

a — bz (a — b)z a — b . a — b , 
x — — • ••• x — a = — , x — b = , dx — - -dz; 

i — z i — z i — z (i — zy ' 

and the expression transforms into 

(I _ z )m+n-2dz 



(a — b)» l + n - l z m 
Expand the numerator by the binomial formula and integrate directly. 

46. Integrate / sin/* cos q xdx, whenever p -f- q is an even negative 

integer. 

Let p -f q = — in. Then 

sinAr cos?* = sinAr cos-/>- 2 «* = tanAr sec 2 "*, 
= tan/*(i -j- tan 2 *)"- 1 sec 2 *. 
Put tan x = t. Then 

/ sin/* cos q xdx = /V(i + t-) n ~^dt. 

Expand by the binomial formula and integrate directly. 

47. Integrate sinAr cos?* dx, whenever / or q is an odd positive integer. 
Let p — ir -f- I, then 

I sin 2r + I * cos?* dx = — f (sin 2 *)'' cos?* ^( cos *), 

= — / (i — cos 2 *) r cos?*</(cos *), 

= - f(i- cycidc. 

Expand by the binomial formula and integrate. 

48. /*sin 3 dB = £ cos 3 — cos 0. 

49. / cos 3 dO = ? Check by putting \it — x for *. 

50. fcos 5 dO = sin — $ sin 3 + £ sin 5 0. 

51. AinSfl cos 7 <# = T \jcos 10 9 - |cos 8 0. 

52. / sin 5 * cos- 2 * dx = sec * -f- 2 cos * — ■£■ cos 8 *. 

53. / |/sin * cos 3 * </* = J sinlr — f sin 2 *. 

54. / cos 3 * esc 5 * dx = 3 sin 3 * — f sin 3 *. 



/3 n 3 1 

esc 2 * sec 3 * dx = | tan 2 * — 2 cor*. 



Art. 134.] METHODS OF INTEGRATION. 191 

56. J sin 2 * sec 6 * dx — \ tan 3 * -f- \ tan 6 *. 

57. / sin** sec** dx — $ tan**. 

58. / cscr sec 2 * dx = 2 tan** (1 -(- £ tan 2 *). 

59. /tan M </0 = /tan"-?© (sec 2 - 1) dO, 

tanw-iO /* 

= / tan*-20 dQ. 

« — 1 J 

cot*0 </0 — — " _ / cot«-20 dQ. 

61. / tan 4 dQ = £tan 3 — tan 6 -f 6. 

62. /*cot 3 </0 = - £ cot 2 - log (sin 6). 

63. fcoffl dO = - £ cot 3 + cot Q + 0. 

64. /*cot 5 </0 = - I cot 4 + $ cof 2 + log (sin 0). 

65. / sin x cos x (a 2 sin 2 * -j- b 2 cos 2 *)V*. 

Note, 4a 2 sin 2 * -f 3 2 cos 2 *) = 2(a 2 — b 2 ) sin * cos * dx. Hence the integral 
is 

(a 2 sin 2 * -f- <£ 2 cos 2 *)*. 



2>{a 2 -b 2 ) 
dx I . lb 



66. /^ 5 X , 2 - 2 = A tan- 1 (- tan *) 

J a 2 cos 2 * -j- b l %\Vl 1 x ab \a J 



Divide the numerator and denominator by cos 2 *. 

/dx 

— : . Divide the numerator and denominator by l/a 2 4- b 2 , 
a sin * -f- b cos x ' ' 

and put tan a = a/b. Then we have 

, f ~ x = _i_ log tan (i* - $a -f lit). 

|/ fl 2 + 32,/ cos (* - a) j^a 2 -\-b 2 ' 

dx 
-j- £ cos *' 

# -f- 3 cos * = a (sin 2 £* -|- cos 2 ^x) -\- b (cos 2 -|* — sin 2 -J-*) 
= (a + £) cos 2 -£* -f- (« — £) sin 2 £*, 
which reduces the integral to the form of Ex. 66. 

Divide the numerator and denominator by cos 2 \x, and put z = tan ■£*. Then 
the integral becomes 



68. f — - 



/dz 
(« + b) + ( 



which is standardized. Hence 

/</* 2 (la — 6x) 
; = — ==- tan- 1 1 . ; tan — V , a > b; 
a -\- b cos x \/d 2 -b 2 (\ a + d 2 ) 

log r ' ' — =— , a < b. 



i^b 2 — a 1 \'b^-a— \/b — a tan \x 



I 92 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 



69. /' *^_ = 1 tan-i 4 + 5 tan ** 

e/ 5 + 4snu 3 3 

•7/1 t . ^ ! + COS X , I I 

70. Integrate / ±- dx = : 

J (x + sin *)<» 2 (jr 4- sin at) 2 " 

71. I x sin x dx = sin x — x cos .*. 

72. f l —?dx = log (1 + *) 2 - *. 

73. /• **** =-1 ^— . 

^ (* 3 + x*y 3 (^3 _j_ x rf 

/dx 
. = log (tan-i*). 
(i -f- x l ) tan-^ & ' 

__ p dx . \x 4- 1 2 — x 

75. / — = 2 sin— 1^ — 2 — = cos-i . 

J VS + 4* - x 2 \ 6 3 

76. f C ° S -V^±dx = sin (log *;. Put * = log z. 

mm p dx 1 . tan \x — 2 

77. / - — : = - log 

J 4 — 5 sin x 3 2 tan |x — I 

78. /" = - tan-i (3 tan x). 

J 5 - 4 COS 2X 3 



CHAPTER XVIII. 

GENERAL INTEGRALS. 

General Forms Directly Integrable. 

135. The Binomial Differentials. — Expressions of the type 

x*{a + bxP)i dx, (A) 

where a, /3, y are any rational numbers, are called binomial differen- 
tials. 

This expression is directly integrable in two cases. 

_ _ TT1 a -\- 1 . 

I. When — — — is a positive integer. 

The substitution is a -f- bx$ = z. Then 



M 



dx = i?^—r 



hence 

(z — a) & zy 
x a (a -j- 5xfi)v dx = ^ ^ dz. 

fib~P~ 

Consequently, when — ~^ — • is a positive integer, the transformed 

expression can be expanded by the binomial formula and immediately 

integrated. 

£f -I- 1 
II. When — - 1- y is a negative integer. 

The substitution is a -f- ^ r/3 = zxfi. 

For, if we substitute x = i/y in the differential x a (a -(- 6xP) y dx, 
it becomes 

— v -?Y-"--2(<zy _j_ by dy, 

which, by I, is integrable when — ^ is a positive integer, 

or, what is the same thing, when 

a -f 1 

193 



194 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cii. XVIII. 

is a negative integer. Also, the transformation a -f- Lv? = z becomes 
b -J- ayP = Z)P. 

Hence, under the transformation, 

x*(a + &**)*<£* = -a fl (3 - 2) V fi Y V 2V2. 

In working examples it is better to make the transformations than 
to use the transformed general formulae, which are too complicated 
to be remembered. 

When a, /J, y do not satisfy the conditions in I, II, the binomial 
differential must be reduced by parts.* 



EXAMPLES. 



i. 



'• h 



x 2 ) 2 
f x*dx 

/* x b dx 

4- f dX . . 
J (a 4- cx 2 f 

x 2 dx 
5. 



n x l dx 

'J (a + ex 2 )* 

/x 3 dx 
(a* + x*f 



/x° dx 
(I + x*fi 

r dx a x 
8. / — s- Ans - n- 

J (I -f* 8 )* (! + *»)* 

9 ' Jx 2 (i +X 4)i 

,, , r i - Ans ' 





1 1 rt 2 lof (a 2 \ 2 \ 




2{a * _ ^j 1 2 1 « 1Q S <* -' )• 


Ans. 


I <7 

1 


4 c\a -f rx 2 ) 2 r bc\a + ^x 2 ) 3 ' 




_ L ] rr C v '2 i t \ 




x 2 + i 4 (x 2 + i)= ' 2 * { * ' Ijl 


Ans. 


X j ex 2 | 




X 3 ^ I or 2 ) 




a\a + rx 2 / 1 3 5(« + ** 2 ) f" 


Ans. 


2tf 2 -f- 3X 2 

3 ( ^ + ^ 2) r 


Ans. 


|(! _|_ ^3)i(^3 _ 2 ). 



Integration of ^-^ 

The substitution is a -f- c.r 2 = a 



136. Integration of {A + c j* {a + c ^ (B) 



o 7 , ^ <& 

CdCr = jsr</.# + zxdz y or 



#2 c — z* 

dx dz 



' ' (A + Cx*)(a + c* 2 )* (^c - Co) - Az v 
which is standardized, being 13 or 14 (§ 126) according as (Ac — Co) /A 
is negative or positive. 

* For formulae of reduction see Appendix, Note 10. 



Art. 137.] GENERAL [NTEGRALS. 195 

If (Ac — Ca)/A = — , the integral is 



, x\/Ca — Ac 



tan- x JI2 



\/A(Ca - Ac) j A{a -f ex 2 ) 

If (Ac — Co)/ A — -(-, the integral is 



1 ^ \/A(a -f- ex*) -f jiy^k - Cfc 



'• At 



2\/A(Ac — Ca) \ A (a -f ex 2 ) — x\/ Ac — Ca 

EXAMPLES 

*£r I x 4/2 
. Ans. — - tan-'— 



2. / A-nc hn-I -> 



/ 

3- /; 



+ X 2 )(I _ X 2 )* x / 2 i / l _ x , 

d. 



Ans. - tan- 



(3 + 4x») (4 - 3x2). 5 4/3 1 / I2 _ gx 2 

dx A 1 , 2(3 + 4*2)4 -f ex 

~^7 — ; — ^T Ans - — lo g , -« 

^ (4 - 3*-)(3 + 4**)' 20 s 2(3 _|_ 4x 2 } i _ ^ 

137. Integration of — / ; qX — *dx. (C) 

a -f- 2 ox -\- ex 2 

This is a particular and simple case of the rational fraction which 
will be treated generally in § 148. On account of its special impor- 
tance we give it separate treatment here. 

Let L represent the linear function p -f- ax. 

Let Q represent the quadratic function a -\- zbx -f- ex 2 . 

I. Consider / — . 

Completing the square in Q, we have 



f d ± = f 

J a -j- 2b x -f- ex 1 J ( 



cdx 



(ex _|_ bf - (b 2 - ac) 
Put ex -\- b ■=. z. Then the integral becomes 

dz 

3 



/. 



(b 2 - ac)' 

This is standardized, and depends on whether b 2 — ac is positive 
or negative. If negative, the roots of the denominator are imaginary 
and the integral is an angle, the standard 13. If positive, the roots of 
the denominator are real and the integral is a logarithm, the standard 
14 (§126). 

If ac > b 2 , 



f 



dx 1 ex -4- b 

IT = / z3 tan ' / a ' (l) 



If ac < 3 2 , 



/ 



^ _ 1 ex -\- b — \/b 2 — ac 

Q 2\/b 2 — ac ex -j- 3 -J- 4/0^ — ac 



196 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 



II. Consider 



fi* 



PL _ pc-qb fdx_ q_ A 
" J Q ' c J Q + 2cJ - 



Since the derivative, Q', of Q is a linear function, we can always 
determine two constants A and B, such that 

L = A + BQ\ 
or p -J- ?•* = -4 -f- 2^i? 4- icBx. 

Equating the constant terms and coefficients of x, 
B — q/2c, A =p — bq/c. 

'dQ 

Q 

The first integral has been reduced in (1), (2), and the second 
is log Q. 

In working examples, carry out the process and do not substitute 
in the general formula. 

EXAMPLES. 

1 f xdx (\ ~ 2 1 I 2X + 4 Iv 

'■ J X* + 4 * + 5 J { (* + 2) 2 + I "^ 2 X* + 4* + 5 f **' 

= - 2 tan-'(^ + 2) + 1 log (x 2 + \x -f 5). 

2. f^-^L = _ _I_ tan-x^ti _j_ 1 log (*a 4. 2 x + 3). 

J^ + 2X+3 ^ |/2 2 gV ^ ^^ 

3. /* — = — ? f- log (* -f 1). 

J x 2 4- 2x + I .* + 1 ^ 5V ^ ; 

4 - / , - fe f "/ ) f r e = 7 lo S ^ + 4-r + 5) - tan-*(* + 2). 

«/ •*" t" 4-*" ~r 5 2 

5 - / ; iT — 2 = - lo ? (3 - ■*)■ 

J 3 + 2X — X 2 

6. f ^fr 1 "^,^^ ^ = 2jt - log (* 2 + 6x + io)* + 11 tan-i(x 4- 3). 
7 - f- *7* y l\ = x - lo § (^ + 2^ + 2)2 4- 3 tan-(x 4- 1). 

j/ JV —J— 2^* *-j— 2 

/i'T.r) 
— - dx, where F{x) is any polynomial in x, divide J\x) 

by Q until the remainder is of the form L/Q, and integrate. 

138. Integration of (/ + fl* j ^ ^ 

|/tf -f- 2&V -|- C.V~ 

Let, as in § 137, L and ^ represent the linear and quadratic 
functions respectively. 



I. Consider 



rdx 
J Q T 

square 

f—= r f 



Complete the square in the quadratic, and then 

dx 



Art. 138.] GENERAL INTEGRALS. 197 

which is the standard 11 or 12 according as b 2 is greater or less than 
ac. If a and c are both negative and ac > b 2 , the function is 
imaginary. 

We have, according as the roots of Q are real or imaginary, 



—7=- log [ex — I — 3 — I — \/c{a 4- 2bx -f- C.* 2 )], 

1 . ex -f- ^ 
— — sin * — — , 
\/c \/ac -j- £ 2 

as the corresponding values of the integral. 



II. Consider 

Write, as in II, § 137, L = A -\- BQ\ and determine A and B. 
Then 



, § 137, L 
al on th< 

r dx 
J LQ? 



The first integral on the right was reduced in I, the second is 

2(2*. 



III. Consider 

J ^0} 

a dx dz 1 — pz 

Put p -f ox = i/z. . ■. — — t = , x = — . 

r ' * 7 p + <i x z q z 

Substitute in the integral and it transforms into 

dz 



f, 



\/a f -j- 2b 'z -\- c'z 2 
which can be integrated by I, then replace z by i/(p -f- ?■*)• 



EXAMPLES. 

f , = 2 log (|/* + V* - «)• 



- C dx . \x . (2x \ 

2. / =^=r = 2 sm-i — = sin- * 1 ) . 

«/ |/a^r - x 2 \ a \ a I 

/dx 

— — . = 2 sin- 1 ^ — l = sin- I (2x — 3). 

^■$x — x 2 — 2 

4. /* , dx = log (2* + 1 + 2^1 4 * 4 * 2 ). 
* |/i 4- ^ 4- x 2 

5. fJ^~dx = */(* + «)(* 4-*) + (0 - 3) log (y*T5 + 1/^M). 

/• <2r 2x 4 I 

6. / — - = sin- 1 — — — . 
J j^-l — x — x 2 V5 



198 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 

_, P dx 1 a 

/• / = - cos- T — . 

J x\/x 2 — d l a x 

8. f dx =-J^- 

9 . f ** = -1**," + **+*. 

J xi/x 2 4- a- a 



10. 



^2 _|_ fl2 a x 

? dx 1 . xi/2 

/ = — - sin- 1 — - — . 

J (I -f x)f/l -|- 2X - ^ 2 4/2 I + * 



|1. f t = ^ i/x>-2X+ 3+i /2 

J (x - l)*/x*- - 2x + 3 |/ 2 X - I 

12. r (* + 3)^ = tfj + 2x+ - + log (x + j + ^ + 2 ^ + 3)2. 

«/ |/x 2 -j- ix -|- 3 

Reduction by Parts. 
139. Integration of Powers of Sine and Cosine. 
/ sin n xdx = 1 sin" -1 * sin * dx. 

Put & = sin" -1 *, dv = sin x dx; 

. t. du — {n — 1) sin" -2 * cos x dx, v = — cos x. 
Hence, applying the formula for parts, 
isin n xdx= — sin* -1 .* cos x -\- (n — 1) / sin" -2 * cos 2 xdx, 

= — sin" -1 * cos x -f- (n — 1) / sin"~\*(i — sin 2 x) dx, 

— — sin" -1 * cos *•-[-(>/— 1) / sin n ~ 2 xdx— (n— 1) / sin M *</*\ 

/sin* -1 .* cos jf , » - 1 /• ', 
sin"*#*= 1 1 sin* -2 .* dx. (1) 

When « is a positive integer this reduces the exponent by 2, and 
leads to / dx or / sin x dx according as n is even or odd. 

Since integration by parts depends only on the differential equa- 
tion d(uv) =. udv -\-v du, the formula is true when n is any positive 
or negative rational number. 

Change n into — n -J- 2 in (1), and we have 

/dx — cos x 7t — 2 P dx 

sin"* ~ (n — 1) sin* -1 * "" n — ij sin" -2 .*' ^ 2 ' 

In (1) and (2) change x into \n — x, then 

/„ , cos" -1 * sin x n — 1 f 
cos*.* dx = -|- / cos" -2 * dx, (3) 

C dx _ sin x 71—2 r dx 

J cos** "~ (n — i)cos* -1 * n — 1 J cos" -2 * ' 



Art. 139.] GENERAL INTEGRALS. 199 

These formula are important. They reduce the integrals to stand- 
ard forms whenever// is an integer. 

Formulae (1), (2), (3), (4) can be obtained directly and in- 
dependently by integration by parts. In practice this is the better 
method. The separation into the parts u and dv is indicated in each 
case in the formulae below. 

/sin" , /'sin" -1 w sin , 

co,» xdx =j cos"-* X cos***' 

f sec ixdx= f sec z xx se ixdx. 

J csc M / csc M 2 esc* 5 

In the part tvdii use sin 2 * -f- cos 2 .* = 1, sec 2 * = 1 + tan 2 jf, 
or esc 2 * = 1 -J- cot 2 .r, as the case requires. 

EXAMPLES. 

/» . sin x cos jc , 

1. / sin 2 .* tf .* = h £•* = : £•* — i sin 2„r. 

2. / sin 3 jr dx = — £ sin 2 x cosjr — f cos x = % cos*.* — cos .#. 

3. I sin 4 .* ^r = — i cos jf sin x (sin 2 jc + I) + f-*"- 

4. I sin 5 .* <s6r = — £ sin** cos x -\- f f sin*r a^r. 

5 . / sin 6 jtr a&; = — £ cos x (£ sin 5 .*- -f T 5 ¥ sin 3 jp + £■ sin x) -f- y 5 ^. 

6. Find the corresponding values for cos x, integrating by parts. Check the 
result by putting £ it — x for x. 

dx 



/dx 
= log tan \x — log (esc x — cot x). 
sin x 

8. / X 9 = - cot jr. 
J sin 2 jc 

/<& COS Jf I JT 
r- = — — r-= — H — log tan — . 
sin 3 jtr 2 sin 2 * ^2 5 2 



^ r dx 1 cos * 2 

10. / -r-j— = ^1— — - cot x. 

J sm*x 3 sm x 3 

/djc 1 cos x 3 cos jr 3 * 

sS7 = - 4 1EST - 8 1ES + 8 log tan r 

<£c 1 cos jc 4 cos * 



'*■/ 



_ — cot X. 



sin°* 



13. Deduce the corresponding integrals of cos x, and check the result by 

putting \tc — x for x. 



200 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 

140. Integration of / sin" 1 .* cos".* dx. 

We have for all positive or negative rational values of m and n 

d sin 7 " - '.* . . sin w_2 .* sin'"* 

— = (m — 1) . L (n — 1) . 

dxcos H ~ l x v ' cos* -2 .* v cos "a - 

Therefore 

/sin"*.* , 1 sin w-1 .* m — 1 /*sin w_2 .* 
dx — — - / dx. (<\ 
cos".* n — i cos n-I :v n — 1 J cos" -2 .* v 

In particular, when m = n, 

r , tan*- 1 * Z 1 

/ tan*.* <£* = — — / tan* -2 .* dx. (6) 

Put \7t — x, for x in (5) and (6). Then 

/cos ws * , , — 1 cos'* -1 ^ 7ii—\ rcos m ~ 2 x , 
</.* = : _ / dx, (n\ 
sin*.* n— ism* -1 .* n — 1 J sin* -2 .* w/ 

r J cot*- 1 * r 

/ cot'"* dx = — — / cot* -2 .* dx. (8) 

The same results are obtained immediately by changing the signs 
of m and n. 

Change the sign of n in (5), then 

/sin** -1 .* cos* +I .* m — 1 f* . 
sin"'.* - cos*.* aL* = ■ 1 / sin" 1-2 ^ cos*+ 2 .* dx. 
n + i ^n+ij 

But sin'" -2 .* cos* +2 .* = sin'" -2 * cos**(i — sin 2 .*), 

= sin"' -2 .* cos** — sin'".* cos**. 

Substituting and solving, we have 

/m — 1 f . sin'" -I .*cos* +I *- . . 
sin'"* cos"* a* = / sin'" -2 * cos"* dx ■ . (o) 
m + nj m + n Ky/ 

In like manner, change the sign of n in (7) and write 1 — cos 2 .* 
for sin 2 * in the last integral. Then 

/ m . H , m—if . . M , . cos w - J ^sin w+I j; 
cos'"* sin** #.* = / cos'" - x sin** dxA . 10 
n + mj m+n v ; 

These formulae serve to integrate sin*.* cos"'.* dx whatever be the 
integers m and n. 

It is well to be able to integrate the functions of this article in- 
dependently. The forms below show the separation into the parts u 
and dv which effect the integration directly when the trigonometrical 
relations sin 2 .* -|- cos 2 .* = 1, sec 2 .* = 1 -j- tan 2 *, csc 2 .* = 1 -f- cot 2 * 

are used in the integral jvdu. 



Art. 141. J GENERAL INTEGRALS. 



201 



j sin m xcos n xdx= J sin" 1 ~*x cos n xxsinxdx= /sin'".vcos , '- , -vxcos-v < /.v, 



/'tan" _, rti 



tan" J tan'' , 

cot—* X cot* *^' 



EXAMPLES. 



1. J cos 2 * sio*xdx = 4 sin * cos *(£ sin** — J 3 sin 9 * — j) 4- fac. 

r dx 1 

2. / -: 5- = (- log tan \x. 

J sin * cos\v cos x 

3- / -1-5 5— = !-=- + - log tan -. 

J sin j j cos 2 x cos x 2 sni-'x 2 & 2 

4. / tan 4 x dx = £ tan 3 jr — tan * -f- *. 

5. / cot 4 .rd* — — l cot 3 * -4- cot jt -f x. 

* P dx 1 

6. / - — — = — • log (sin *). 

J tan ! * 2 tan 2 * & v ' 

/» dx — 1 1 

7 - / : — r- = — : — 1- + 5- + log (sin *). 

J tan°.r 4 tan*r ' 2 tan 2 * ' & v ' 



Integration of Rational Functions. 

141. General Statement. — Any rational function of x whose 
numerator is a polynomial N and denominator a polynomial D can 
by division be decomposed into 

D V ^ D' 

where Q is a polynomial, and the degree of R is that of D less 1. 
We then have 



f^dx =fQJx +f^dx. 



The first integral on the right can be written out directly. The 
second integral demands our attention. We know from the theory 
of equations (C. Smith's Algebra, § 436) that every polynomial in x 
of degree n has n roots, real or imaginary, and can be written 

A{x — a^)(x — a 2 )...(x — a n ). 

If there is no second root equal to a x , then a^ is said to be a 
single root. If, however, there is another root equal to a lt say 
a 2 = a x , then the two factors can be written (x — a x ) 2 , and we say 
that a x is a double root, or that the polynomial has two equal roots. 
In like manner, if there are r equal roots equal to a, the correspond- 
ing factor is (x — a) r , and we say that a is a multiple root of order 
r, or the polynomial has r equal roots of value a. 



202 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 

Again, we know that if the coefficients in the polynomial are all 
real, then imaginary roots must occur in conjugate pairs (C. Smith, 
Algebra, § 446). Therefore, if there is an imaginary root a -\- bV — 1, 
there must, be another a — bV — 1. Now the product of the factors 
corresponding to these two roots is 



( A - _ a _ bV - i)(x — a + bV — 1) = (.v - af -f b 2 , 

— x 2 — 2 ax -|- a 2 -f- b' 2 , 
which can be written = x 2 -j- px -f- q. 

Moreover, if a -J- & (1 = ^ — 1) is a multiple root of order r, so 
also is a — ib, and we have the corresponding factor in the 
polynomial 

(x 2 +px +<?)'. 

Hence any polynomial in x is composed of factors, linear and 
quadratic, of the types 

x — a, (x — b) r , x 2 -\- px -j- q, (x 2 -\- px -f- q) r . 

if ^M 

A*Y 

be a rational function, in which F(x) is of a degree at least 1 lower 
than that of f(x), we can always decompose the function into the 
sum of partial fractions corresponding to the roots oif(x), as follows: 
For each single real root a there is a fraction 



for each multiple real root b of^order r there are r fractions 

_^L_ + *, ' , + £ r , 

(* _ b) T- ^ _ b y T • • • T ^. _ ^,> 

for each pair of conjugate imaginary roots there is a fraction 

C + Z?-v 

for each pair of conjugate multiple imaginary roots of order s there 
are s fractions of the types 

^, + ^1 , ^, + ^, , 1 E, + xF s 

x* _j_ „.* _|_ ^ "T (^2 _j_ ^ _j_ ^s T • • • -r ( v > + a v + py 

In these partial fractions the numbers A, B, C, D, F, F, etc., 
are constants. Since there are exactly as many of these constants as 
there are roots oif(x), they are n in number. 

If now we equate F(x)/f(x) to the sum of the partial fractions 
and multiply the equation through by /(.v), we shall have F(x) 
equal to a polynomial in x of degree n — 1. When we equate the 
constant terms and the coefficients of like powers of .1* on each side 



Art. 141.J GENERAL [NTEGRALS. 



103 



of this equation, we have // linear equations in the constants A, />, 
C, etc., which serve to determine their valui 

The integral of the rational function then depends on 

r ix and r { E+ Fx)dx 
J (■*-«)' J (**+& + & 

The first of these can be integrated immediately, the second is 
always of the type 

f {E + xF)dx f dz /• zdz 

J u* -v +■*]'" { + 'J W+W + J W+W' 

wherein x = a-\-z. The last integral on the right is 

r *** = * r A z 3 = _±_ -1 
J (« 2 +'« , ) r *J (rf+jy 2 (r-i)(^+^- 1 - 

To integrate the first integral on the right, j put 2 = b tan 6. 
.-. dz = dsec 2 6d6. 

Then 7wTn == ^f C0S ' r " e ' ls ' 

which can always be integrated by parts, § 139. 

Hence the rational function can always be integrated. 



EXAMPLES. 

J x A — 4jc 
We have here single real roots; hence 



x i + 6x - 8 x 2 + 6x - 8 ^ ^ C 



.* a — 4* x(x — 2)(x + 2) # ' JC — 2 ' X + 2 ' 

Clearing of fractions, 

x t + 6x - 8 = A{x - 2)(x + 2) -f- ^(x + 2) + C(* - 2)*, (1) 

= (A -f j9 -f- C)x 2 -f 2(^5 - C)x - \A. 
Equating coefficients, 

A + B -|- C = 1, 2(^ - C) = 6, - 4^ = - 8. 
.-. A = 2, j5 = 1, C = — 2. 
Hence the integral is 
-2 _j_ 6x - E 
~~$x~ 

. X\X - 2) 

= l08 r . 

g (X + 2) 2 

If we assign particular values to a - in (1), we can find A, B, C 
more easily. Thus put x = o, then — 4^ = — 8; put x = 2, then 

* Provided these « equations are independent, which they are. 
-j- See also Ex. 88, at the end of the chapter. 



/ x 2 _|_ d x _ £ 
^3 _ ^ ^ = 2 log x + log (x - 2) - 2 log (x + 2), 



204 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 

SB = 8; put x = — 2, then SC = — 16, which give the constants at 
once. The general principle involved in this abbreviated process is: 
when there are only single roots, put x equal to each root in turn, and 
the constants are immediately determined. 

2 - f i iX Tu X) ^ o\ = t log (* - 3) + I log (x + 2). 

J (x - 3){* -T 2 ) 

3- f . *i*. , = ! log (* + 3) + i log (x - i). 

4 - Ix^Vx^ix ^ = rJogx + l log (x - 2) + i log (, + 3 ). 

6 - f x * 3 lx-6 d * = log [( * + 3)2 (X ~ 2)L 

7. T X<dX = — - 2X + ' log * ~ * + -V- I©? (* + 2). 

/~.3 I J 
±- — - dx. Here there is one single root, o, and a triple root, x = I 
.*■(.* — l) 3 

Hence 

* 3 + i _ A B C D 

x{x _ i)3 x -r (* _ !)3 -r ^ _ I)2 1- ^ _ x ■ 

Clearing of fractions, we have 

x % -f i = (A + Z>)* 3 + (C - 3^ - 2i?)^ 2 -{- (sA -\- B - C + D)x - A. 
.: I = A + D, 

o = C - $A - 2D, 
o = 3A + B-C + £>, 
i = - A. 
Whence A = - i, .5 = 2, C = I, Z> = 2. 

•T 3 + I _ _ I _J 2 , * , 2_ 

x(x — I) 3 x "*" (# — I) 3 "^ (Jf - I) 2 "" jr - I ' 

/Jtr 3 -(- I II 
— — dx = — log x — ■ s (- 2 log (x — I), 
*(.*• — i) 3 (x — i) 2 x — i [ 

(x - i) 2 



= log 



X (x - I) 2 



— 7-t— = . Here there are a pair of imaginary roots. 

(x + i)(x z + I) 

x A Lx + J / 

(7+~l)(** + I) - ! + , + I -t «■ ' 



Art. 141.] GENERAL INTEGRALS. 205 

Clearing of fractions, 

x = A(l + x 2 ) + [Lx + M)(l + *), 

= (A + J/) + (Z + J7> + (^ + ^)-v 2 . 
Equating coefficients, 

Z -f A = o, Z -f M — I, ^ -f J/ = o. 
Z = {, J/ = i , ^ = _ \, 

r x dx 1, I + x 2 1 

••• J (T+ ix,- +1) = 4 ,og (TT^F + 2 tan " 1 " 
13 - /rq^ We have * + ■** = (I + r)(1 - * + x ' l) - 

1 ^ Zx + M 

■"'- I +X 3 "~ T+^r + I - a- + x 2 ' 

Clear the fractions and put * = — 1. Then A ~ \. Substituting this, we 
get 

2,{Lx -f M) = 2 — x. 

/dx _ 1 /» <&■ 1 /* (2 — *)<&■ 

I + jt 3 ~ 3«/l + A:~'3 t /l— jr-f-x 2 ' 

II I 2x I 

log (i + x) — -j: log (I - x + x 2 ) 4 ^tan-i 



3 6 V3 y 3 

14. f * = I log - I + " + " 2 + —tan- 2 " + ' 

e/ I - X 3 6 g I - 2X -f- X 2 ^ |/7 

15 f x2dx 



1/3 v 3 

- 2 </x 



A B Lx + M 



(x — if(x 2 + 1) [x — i) 2 ' x — I ' 14- X 2 

x 2 = ^(1 4- x 2 ) + Z^x - I)(x 2 4- I) 4- (Lx 4- ^)(jc - 1)2. 

Equating coefficients, 

^ _ ^ 4. M — o, 

Z^ 4- Z = o. 
£ 4- Z - 2 JZ = o. 
. •. M= o, B = A = i, Z = - -|, 
Hence the integral is 

_ 7x-=T+J log(x ~ I} ~ 4 Iog {x ' 2 + lX 
16. /\^ + I^ = 3 log — - + *-tan--. 

r 2 - 3* - 3)^ , (* 2 ~ 2 * + 5) ? 



17- f/ 3i*~ 3 T, = log ^ ~ 2 " +5) + i tan- 

J (X — l)(x 2 — 2X 4- 5) fa X — I '2 

,8 - /f+3?* = ' + 6^-p - V3 tan- j=. 

f ^ — 1 X4 I 

' J (x 2 4- i)(x 2 + x^) _ 4 ° g (x+ l» 2 (.r 2 + I) ~ 2 t<in 



206 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 

/2x^ -)- X -4- "? 
— -^ dx. Here there is a double pair of imaginary roots. Hence 
(x -\- I)* 

we put 

2x3 + ^ + 3 _Ax + B Cx + D 
(x 2 + i) 2 ~~ (jc 2 + i) 2 + x 2 -f I 
. •. 2JK 3 + ^ -f 3 = Cx 3 + Z>x 2 + (^4-^ + ^ + 2?. 
.-. ^ = — i 5 B = 3, C = 2. Z> = o. 



2X 3 



_|_ x _j_ 3 -x+3 , 2x 



1 + 



(x 2 -f- i) 2 (x 2 -f i) 2 ' x 2 4- i 



/a* 
2 — j j P ut x — tan Q> tnen * ne integral becomes 

/cos 2 a?9 = 10 4- i sin 20 = -1 tan-*# -I . 
1 2(jH + I) 

- /^^-=^^)+t^- T -Io g( , 2+r) . 
21. [J*Lt*L* m = ,.; 3 "- 24 _, + -^ tan- 2 ^- 3 



3 X +3V 3(* 2 - 3-*" + 3) 3 |/ 3 4/3" 

22. ffi + J-ZJ c/x = ■ 2 ~ * ; 4- log (^ + 2)* 

,/ (x 2 + 2) 2 4 (x 2 + 2) ^ & v ' ; 



4 y 2 4/2 

142. Trigonometric Transformations. — On account of the simple 
character of the reduction formulae in §§ 139, 140, it is often advan- 
tageous to transform many algebraic integrals to these forms, and con- 
versely many trigonometrical formulae can be transformed into useful 
algebraic forms. * 

EXAMPLES. 
1. Put x = a tan 6, then 

/x vi dx r 

_ a m-n+i / s i n w0 cos w_ w - 2 fa 
(a 2 4- x' 2 ) in J 

1 0, then 

/x™ dx , C 

(«2 _ x 2f« J 



2. Put x = a sin 0, then 
x'" rt'.r * sin 

cos"-^ 

3. Put x = a sec 0, then 
x»' dlr r cos M - w - 2 



/x'» rfx /* ^. 
— a m - M +i I , dQ. 
( A 2 — a 2 f- n J sin"-'0 



4. Put x = 2a sin 2 0, then 

/X™ dx /•sin 2 "'-"+ ] ,„ 

(2ax — x 2 ) in J cos*-'0 

5. Make the same transformations in the above integrals when m or n is 
negative. 



* The reduction formulae for the binomial differentials are given in the Ap- 
pendix, Note 10. 



Art 144. J GENERAL INTEGRALS. 207 

The general integral 



A 



x m dx 



can always be transformed to the trigonometric integral when the signs 
of a and c are known, whatever be the signs of m and n. 

EXAMPLES. 

1 . Integrate by trigonometrical transformations 

f \/a* — x 2 dx, f \/x l — a 2 dx, f \/x 2 + d l dx, 

/dx r dx /* dx 

\Zd 2 — x 2 ' J \/x 2 - a 2 ' J \/x 2 -f a 2 ' 

Rationalization. 

143. Integration of Monomials. — If an algebraic function con- 
tains fractional powers of the variable x, it can be made rational by 
the substitution x = z n , where n is the least common multiple of the 
denominators of the several fractional powers. 



J i-\-x i 



For example, 

+ 
Put x = s 4 . The transformed integral is 

: 3 (i -f z) dz 



* 



I + z 2 
Consequently the integral is 

|jc* — 2x1 — ^xi -\- 4 tan- 1 ^ — 2 log (1 + x*). 
Again, any algebraic function containing integral powers of x along 
with fractional powers of a linear function a 4- dx can be ration- 
alized by the transformation a -f bx = z n , in the same way as above. 

EXAMPLES. 

/x^ dx 2 

—=== tt (5* 3 + t* 2 + 8x + 16) s/x - 1. 
\/x — 1 35 

P x dx 2 2a -\- bx , 2 

2. / f3 = T .._T._.. , by a + £x = 2 2 . 

J {a + bx> P ^ a + ^ 

Complete the differential, integrate and compare results. 



dx , / . n 2 2 yx — 1 -\- 1 

= \og(x -f |/x - 1) = tan— 



+ V* ~ * v 3 v 3 



r (x 2 ± i)dx _ r _ 

J x \/x* -f- ax 2 -f~i ~ ^ 4/z 2 + 



Put x T x-i 



± 2 



144. Observations on Integration. — As we have remarked be- 
fore, comparatively few functions have primitives which can be 
expressed in a finite form of the elementary functions. For example, 



2o8 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 

/ \/y dx, when y is a polynomial in x of degree higher than the second, 

is not, in general, an elementary function and cannot be expressed in 
finite form in terms of the elementary functions. \iy is of the third 
or fourth degree, the integral defines a new class of functions called 
elliptic functions. 

Functions that are non-integrable in terms of the elementary 
functions can frequently be expanded by Taylor's series and the integral 
evaluated by means of the infinite series. 

Any rational algebraic function of x and \/ax 2 -\- bx -j- c can be 
rationalized and integrated as follows : 

Factor out the coefficient of x 2 and \ety = \/±x' i -\- px -j- q. 

The rational function F{x,y) is rationalized in x: 

I. When the coefficient of x 2 in y is positive, by the substitution 

\/x 2 4- px -\- g = z — x. 

Then , = Z -Z± , s _ x = t±*±l, dx= ^±J*+fi liz . 

p -f- 2Z p -f- 2Z {p -j- 2Zy 

J ^ ' J \P+2Z' P+2Z )(pJf2Zf 

II. When the coefficient of x 2 is negative and the roots of the 
quadratic a, (5 are real, then 

— x 2 -j- px -j- g = {x — a)(fi — *)■ 
The function F(x,y) is rationalized by either of the substitutions 
\/ — x 2 4- p x -\-q = \/{x — a)((3 — x) = (x — a)z or (/? — x)z. 

]dz, (*_«), = !£= 

-f- (3 (ft — a)z\ z dz 



T hen x = - — ~- , dx = — - v — — — ' dz, (x — a)z = y - — - ; . 

i 4- z l (i 4- z 2 ) 2 v i -j- z 2 



.-. fF( X ,y ¥x = 2{ a-P)fF(^- + 1 + ^ /(M 

When the roots of — x 2 -\~ px -\- q are imaginary the radical is 
imaginary. 

145. Integration by Infinite Series. — We know that if a function 

A x ) = a o + a i x + V* + • • • 
in an interval ) — H, -\- H {, then also its primitive is equal to the 
primitive of the series for this same interval (§ 72). Hence 

J/(x)dx = a Q x + \a x x 2 -f- \a 2 x* + . . . 

EXAMPLES. 

f dx - - 4- i - 4- 111 x " 4- * 3 5 ^- 4- 
' «/ "4/! _ x 5 _ I 2 6 2.4 11 "^ 2.4.6 16 ' 

2. / = 2 4/sin -r 1 -f • + . . • ) 

J |/sin.r V 2 5 ^2.4 9 / 



Art. 145.] GENERAL INTEGRALS. 209 

Put sin x = z. .-. i/x — dz/cos x, and the integral is 

r * 

J \w q m 4- n ' 2 ! </ 2 #» 4- 2» ' / 

For what values of x is this true? 
4. Show that 

dx x 1 x 5 I • 3 x 9 



I -J 
^ 4/1 



a: 2 < I. 



jr» 1 2 5 2.4 9 

1 11 i-3 1 

__ _4_ t r _i_ 

x ' 2 5x 5 2-4 gx 9 ' 

5. Show that 

J *-+** = ' \ log( 3 + ^ + r-i- + 2-! L ^- + 

Determine the values of x for which this is true. 

Put b -\- x — z. .-. e ax == e- ab e* z , etc. 

6. The elliptic integral / (I - £ 2 sin 2 *)**/.*, £ 2 < 1, can always be ex- 
panded by the binomial formula, and the general term I sin 2 ** dx integrated. 

., /'sinx , f/l ix 2 ix 4 \ 

7. / — — dx = 2x I — 4- , — . . . ) . 

J |/x \3 7 3! T "5! / 



EXERCISES. 

x* dx 



,— — - rr = f Sin-x* ~ ** Vl%- * 2 (3 + 2^ 2 ). 
(i — X 2 )* 



/x I — 4/1 — X 2 4/1 — 
=^log — 

I— X 2 

dx x 



2. /_^=. = i log 

J x* V I— x* 



4/ t -* 2X 2 



3. f dX = - 

V ( fl 2 _[_ X 2)| fl4 ( fl 2 + X 2)i 3^4(^2 _J_ ^f 

. /» x* dx — X s 3/ x X ■- 

4. / = = r 4- - Ix — a tan-i _ ) . 

J {a* 4- x 2 ) 2 2{a 1 4- * 2 ) 2 \ <z / 

47 (2#x — x 2 )* ^^« 

x 3 e*x dx = — x 3 - -x 2 -I -x r) . 

\ a a 2 a 3 / 

7. /"x 3 (log x) 2 dx = ix* [(log x) 2 - * log * 4- £]• 

8. / x 3 cos x dx = x 3 sin x 4- 3^ 2 cos x — 6x sin x — 6 cos x. 

9. /x 4 sin x ^/x = — x 4 cos x 4- 4-r 3 sin x 4- I2x 2 cos x — 24 (x sin x 4- cos x). 



sin 2 fl ^ 
(1 4- cos G) 2 



10. / , _^i J 

e/ (I 



2io PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 

11. /cos 3 G sin 20 dQ = — § cos 5 0. 

12. / sin 2 cos 3 dO = § sin 3 — \ sin 5 0. 

13. / sin 5 cos 5 dQ = — — (cos 20 — § cos 3 20 -f £ cos 5 20). 

14. / cos 4 .* - esc x dx = i cos 3 .r -f - cos .r -f- log tan $x. 

15. f cos 4 * csc 3 :r dx = (cos 3 .* — |- cos .#) csc 2 .# — f log tan \x. 

16. f * = — 1= log ^4+^-^"^!. 

17 (2 + 34(4 - -r 2 )* 4 1/2 4/4 + 2jr -)- 4/2 - j; 

r - (t 8 ^ + t* 2 + iwi + ^r 1 - 

(I -f- x' 2 )- 
18. /* ^t r. Put 2 = (tf 2 + **)* + X. 

Show that / j: w [(a 2 -f- x" 2 )^ 4- ^]"</r can be integrated by the same sub- 
stitution when mis a. positive integer. 

•/ (I + Jf 2 )* * 

20. /-^^i^-ilog^+i). 
J x* -f 1 3 3 

J x s _L *3 ^5 * 



25. 



f^+A dx = - 4 + 4 + log * 2 - 24 log (** + 1). 

f -T- 1 '—, = -* f + 2 1°S , ~ * + 4 tan-***. 

J jch — x B 3 jt* -(- I 

/• * =fc g * /r + 1 -'. 

«/ Jf|/jC +1 |/JT + I + I 

/• x 2 dx _ 6x* + 9^+1 
./ (4* + i) §_ 12(4* + 1)3 
<*r 3 



. r ox_ 

tm Ji + yi 



26. I %= = & + i)* - 3(* + i) 1 + 3 log (1 + f 1 + *)' 



+ * 



27. I^-* =&*_••)(* + .)*. 

«/ (** -f a)* 3 

28. f-?JLz = -lj/T^~^^ 4- 2). 

•/ 4/1 - * 2 3 

29. f ***- = ^(3^ - 2* 2 + 2)4^2^ + 1. 

J \/2X 2 + I 3° 



Art. 145.] 



GENERAL INTEGRALS. 



211 



30. f**(a* - **)* d* = —{6x* - <i 2 x- - 5«*)(« 2 - x 2 )K 



dx 



I , V'a- 4- a 1 — tf 

— . = — log; ! 

x tfx* 4_ a 2 2a ^ 4 / -v- + a- -\- a 

ix 



dx _ I . \?2 + 2X — 4/2 — X 

2 + X — X 2 |/2 ^2 -f- 2X -j- |/ 2 — *' 



32. / -: r "! = log (4/J315+,)* 

•'■**.+ 24/3 -«" 

33. /*— 



+ ^g (4/3 



3)*. 



«/.l- 



log 



|/X 2 _x-f-2 + X— |/2 



35 



36 



x\/x 2 — jf + 2 4/2 V-* 2 — * + 2 + * + V2 

dx 



X]/x 2 + 2X - L 



2tan- I (x -j- yV 2 -f- 2* — 1). 



. f^±-^ ** = -* ^ + .o g ( ^m + V?)' 

__ /* |/6jc — x 2 . ,6 — x 

37. _/ ^— * = - 2 ^j-— + 2 tan-^ 

f dX 

J (X — I) 2 \/x 2 — 2X -j- 2 



38 



39 



4/.V 2 — 2jc -4- 2 



(*-2) 



3*- 5 



(* - 2) 2 ' 



mmk r X s dx 

«■ y^+i? = ? 

4,. /• ^= = ilog *—= 

J x i/a' ± x 1 a a+ fV ± x* 



dx 

x 3 dx 



Put X — 2 = Z. 
Put JP + I = z. 

Put xz = a. 



42. f f. , = f (x 2 - 3)(* 2 + I)*- 

J (x 2 + I) 3 



Put x 2 -\- 1 — z. 

4-3. / . n * "* ; = (x-\-a) cos <z — sin a log sin (x 4- a). Put x 4- a = z. 
J sin (jc 4- a) v • ' 



45. 



/ 



sin j; </.r 
n (j?4 

dx 



/ e 2X dx 
. = ^3** - 4)(** + 0* 
(f+iF 



Put e* 4- I = z. 



e 2X _ 2 tf* 2<?* 4 4 



- + - log [e* - 2). 



46. /V 2 \o%x dx = \x* (log x - i). 

47. / x*- 1 log x <** = — x n (log x \ 

48. f x sin x dx = — x cos jc 4" snl •*"• 

49. /*.* log (x 4- 2) </* = (* 2 - 4) log 4/7+2 - \x* 4- 



212 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 

50. / x tan-ijr: dx — \ (x l 4- 1 ) tan— *x — lx. 

51. Integrate 

fx-*{cP - x*)-* dx, f(a 2 - x 2 )* dx, f ' x* (a* 4- x 2 )-* dx, 

tx* ^Tp dx, fx* */a* + x*dx, f(a* - x^f dx. . 

52. I sin*.* cos 2 jc dx = •£ cos jc(£ sin 5 * — T x 5 sin 3 .* — £ sin .r) -}- -£ s x» 

- n f dx I , tan lx 4- 2 

53. / — ■ = - log 

J 3 -f- 5 cos x 4 

_. ■ /• dx i 

54. / = — tan-i(2tan \x) 

J $ — 3 cos .r 2 



3 -)- 5 cos x 4 tan -kr — 2 

5—3 COS X 2 

— — - = - . Put XZ = I. 

{a -f rx 2 ) 2 fl(fl 4- ^c 2 ) 2 

56. /• dx *-±^ ,. 

J (a 4- 2 for 4- rx 2 ) 2 (a* - £ 2 )(a 4- 2&* 4- cx 2 f 

Complete the square and put ex 4- b = z. The integral reduces to 55. 

5? f (P + qx) dx = fy- aq 4- (cp - bq)x 

* {a -\- 2bx 4- cx 2 )l [ac — b 2 )(a 4- 2bx 4- cx*f 

For #2 = I transforms 

x dx . — dz 
- mto -. 



(a 4- 2bx 4- cc 2 ) 2 {az l 4- 2^2 4- o* 

x dx a -\- bx 



/; 



(« 4- 2bx 4- rjr 2 ) 2 (<w - b*)(a 4- 2 A* 4- rx 2 / 

Combining with Ex. 56. the result follows at once. 



58 



* (3 4- x) dx 7^ — 4 

(1 — 2x4- 2x-)* (1 — 2x 4- 2jr 2 ) 2 



•/^ 



/* c£r I ' (■* — «)(x — £) 

J pT- «)(* - £)(2x - a - b) = (a - by ° g (2x -a - bf" 
r dx 1 . (x — ar(x — b) 

" J (x - a)(x - b)($x -2a -6) = 2{a - bf ° g (3* - 2a - bf 

61. f ,— ^ 8 = i log <« - ** " 3) - 

«/ .r 3 — dx 1 4- 1 ix — 6 2 & 



6x 2 4- ii-r — 6 2 & (jt — 2)* 
__ C x % a„ 

62 - ./ (,-,)(.,- — 



(.r — l)(.v - 2i(x — 3) ~2 & (x — 2) 16 



63. /• , " -, = ±V* «- '** + *> 

J X s — Jx -\- 6 20 (.r — i) 5 

ra f x ' idx _ _ I r (3^ 2 - 7 4-7)^ 
Jx 5 - 7jc 4- 6 " _ 3 J x* - 7x 4- 6 ' 



= l 1 „ g( .^ 7 , + 6) + | )1( , s (--^ + 3) 



Art. 145.] GENERAL INTEGRALS. 213 

67. / , - F *L_ = _i + I 1 lo B '_*_. 
J jc*(0 — x) ax a i a — x 

68 - J ( — - ov - 2) = J3i + 2 lo s jri- 

/</.* _ 2 1 ' r — * J 2x — a — b 

{x - af (* ,- £) 2 = (a - £) 3 ° g x37z " (a - £) 2 (x - fl)(jr - £)' 

70. / -=7-5-: — 2\ = j- + -. cot-i -. 

J x 2 (x 2 -\- a 2 ) a 2 x a z a 

<** _ -1 , ,.,„ (** + 2X + 3) ,,> , ' ta D --* +I 



7, -/i^+- 3 = 67^T) + l0g 



(X - I)* 18 |/2 ^2 

Notice x* — 4x-(-3 = (jc— i) 2 (x 2 -f 2.r -f 3). 

72 - / (*-.)»(£-»,+ ,) =^1 + '° g j.-Zr+« + ' tan "' ( - r - "' 



3 ' J (x 2 + a 2 ) 2 ~ 2T 3 an ' a + 2a 2 x 2 + a 2 ' 

74 /* ^ Jw ^. l 1 ! , 

J x(i + x 3 ) 2 3 g i + ^ r 3i + x 3 



Put X 3 = 2. 



75. / = — log — — 5T — tan- 1 .*-. 

J x- K i + x + x 2 -f- x 3 ) 4 g '* ' 



i/x _ 1 x* I 

X 2 -f- X 3 ) ~ 4 ° g (I + X)\l + X 2 ) ~ 2 



76. / ; , , „ -- ; — = - tan- 



dx _ 1 _ x 3 + 3* 

X 2 d£t I 2X 3 

= -7- tan- " 



77. /* X ' ldx 

- a f x>dx 1 x 2 « 



79. If/(jc) = (* - atj) . . '. (x - a M ), and T^x) is a polynomial of degree less 
than n, show that 



/ --}-{ dx = 2 7T - L . log (x - a r ). 



80. Show that any algebraic function involving integral powers of x and frac- 
tional powers of 

a -4- bx 
y = p+qx 

can be rationalized by putting y = z m , where m is the least common multiple of 
the denominators of the fractional powers. Apply to Exs. 81, 82. 



a/x — b -f- \/x — a 

g V — i / — =•• 

|/x — b — yx — a 



fJ^ZT^ = |/(* - a)(x -b)- ^(a - b) log 



214 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 

83. If f(x) is a rational function of sin x, cos x, then f(x) dx is rationalized 
by the substitution tan \x = z. 

-,, . 2z i — z 2 2.dz 
1 hen sin x = , cos x — — , , ax = -. 

I -j- Z 2 ' I + Z 2 ' I -f z 2 

In particular, when m — I, n — I, or m ~\- n is even, say 2.r, we get for these 
respective cases 

s\n m x cos n xdx = — c w (i — c l ) r dc, 

= -f- s™(i — s 2 ) r ds, 

Pn dt 

where j = sin x, r = cos .*•, / = tan x. 

/dx 
-, where Q x , Q 2 are any quadratic functions of x. 
Q1Q2 
Write out Q x ~ 1 in partial fractions. This reduces the integral to § 136, (B), or 
to § 138, (D). 

85. In general, if /(x, y) is any rational function of x and y, where 

y 2 — a -f- 2bx -f- ex 2 = c(x — a)(x — ft), 
then any one of the following substitutions will rationalize f(x, y)dx: 
y = a$ 4- xz, 
=-. z + xc\ 
= z(x — a)\/c. 

f dx _ C x ~ %dx _ 2m+3 Put u — x~* n +~, 

J (a 2 4- x 2 ) n ~~ J (1 4- a 2 x—' i ) n ' X * ' dv ^ or * ne other factor. 

r dx _ 1 _ \ ^ x _ J* dx 1 

J (a 2 4- .r 2 )« — 20 V - 1) J (tf 2 4- * 2 )»-i + ^" ~ 3 V (<z 2 +.r 2 /<-i f * 
87 . Given the signs of the constants a and b, transform the binomial differential 
x a (a 4- bjfiy dx 
into the trigonometrical differential 

C s\n m x cos"jt dx, 
determining the constants C, m and n for each case. 



CHAPTER XIX. 
ON DEFINITE INTEGRATION. 

146. The Symbol of Substitution. — We use the symbol 
Fix)]**' 

x = a 

or, in the abbreviated form when the variable is understood, 

a 

to mean that the number a is to be substituted for x in the function 
and the result subtracted from the value of the function when b is sub- 
stituted for x. Thus 

F(x)\ B F{t) - F(a). 

If F(x) is a primitive off(x), then we have 

[fit) dt = F(t)]= Fix) - Fia). 

The definite integral is a function of its limits. If one limit is 
constant the definite integral is a function of one variable, the other 
limit. 



147. Interchange of Limits. 

Since f A x ) dx = F ( 3 ) - F ( a )> 

f b /(x) dx = F{a) - F(b), 
•*■ C A x ) dx- - C f{x) dx. 



That is, interchange of the limits is equivalent to a change of sign 
of the definite integral. 

This is also at once obvious from the original definition of an 
integral. For dx has opposite signs in the two limit-sums 

j a f{x) dx and f h A x ) dx > 

while they are equal in absolute value. 

215 



216 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cii. XIX. 

148. New Limits for Change of Variable. — If we transform 
the integral 

$*f(x)dx 

by the substitution of a new variable for x, then we have to find the 
corresponding new limits. 

Let the substitution be x = <p(z), which solved for z gives 
z— ip(x). Then, when x = x , we have z = i/:(x ), and when x = X, 
Z=tp(X). Also, 

f(x)dx =/[0(«)]0'(«) dz = F(z) dz. 
.-. f X /(x)dx = f Z F(z)dz. 

X 

For example, put x = a tan z. Whence z = tan— J— . When x = o, then z =0; 

a 

when jt = a, then 2 = ^7T. Consequently 



r dx _i_ /•** 

^0 (a 2 + X 2 } § ~ « 2 J 



cos z dz = = 

a 2 f/2 



since / cos z dz = sin 2. 



149. Decomposition of the Definite Integral Limits. 

If 

f A /(x)dx = F(X)-F(x ), 



then 

x 

f X /(x)dx = F(X) -F(a). 

Whence, on addition, 

f" Ape) dx + f'"/(x) dx = C X /(x) dx. 

Therefore a definite integral is equal to the sum of the definite 
integrals taken over the partial intervals. This is also immediately 
evident from the definition of the definite integral. 

EXAMPLES. 

Evaluate the following definite integrals: 
jr i_ ]3 27 1 26 



1. / x*dx = -\= J.-- = 
J\ 3J1 3 3 

2. / — = log x I = log e — log 1 = I. 

r.\lt Ml 

3. / sin x dx = I 

Jo Jo 



dx — 1. 



Art. 149.] ON DEFINITE INTEGRATION. 217 

{ P X _ xtidx _ 4/*. 5. / -v ^ = 1. 

— ■ — 5 = log i/2 . 7. / -— 5 = 27ffl 2 . 

a 1 + x' & T Jo x* 4- 4" 2 

~ *r = * Q /•^ vers , r/0 = ^ 

Jo f/a*-** 2 Jo 

10- /***sin 2 dB = \n. 11. / "cos 29 dO = 4. 

12. f k *cos*x sinxdx=i. 13. ^ (| |/T - & fl)dt = 2 yj- 5. 

U r-^-=— 15. re-**dx=L.' 

' Jo a ' 1 + * 2 2a ^° <* 

7T 

16 . f* _*!_ = ♦ . ,7. /* T ^L = log ' + j^ . 

(i 



■ J I -f 2* COS -f- x 2 2 J I 



18 

-j- 2x cos -(- -** 2 sin 



19 



•J 
m /»oo « 



X* 77Z /»oo 

*>-<** sin wjc tf.* = -5— -. 20. / c - ax cos mx dx _ 

a 2 + w 2 J a 2 + 0* 

/ += ° dx * u 

— ; 7 ; 9 = ■ — , when ac > <r. 

22. Show, by putting x — l — z, that 

ixt-Hl — x)l~^dx = fxi-^i - xy-^dx. 

This is called the First Eulerian Integral. Integrating by parts, 

f x *-i(l - x )m-i dx = ** (l ~ n X)m ~ l + ; ^^fx»(l - x)>"-* dx. 

Use this to show that the value of the above integral is 

Jo PKP + i) .-.(/ + * - 

when ^ is a positive integer, and therefore whenever / or ^ is a positive integer 
the integral can be evaluated. 

23W. f**li — x$dx = . 23( 2 ). [x*(i-x)l<tx = . 

Jo ^ 3-7-H-I3 Jo 5-7.9-13.17 

24. The integral / ^— x x n dx is called the Second Eulerian Integral or the 

Gamma-function, T{n -j- I). 
We have, by parts, 

e~ x x n dx = — <r-*r M -f~ w / ^ _ *^ ,n—I <£*. 

Since r - *.** = o when .# = o and when .r = 00 , 

J^ r-*r w dx = n j e— x x n — 1 dx. 
Jo 

.-. r(»+ I) = »J». 



2 1 8 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cii. XIX. 

Also, when n is an integer, 

r(n + I) = n\ 
The Eulerian Integrals are fundamental in the theory of definite integrals. 

25. £ .-"(log *)- ** = £l2g_ = (-.)- jf '*« (log I) "*. 

Hint. Put r~* = a: in Ex. 24. 

J/* 00 » ! 

I ^-azz* dz = — — - . Put x — az in Ex. 24. 

a* +I 

I cos"* <£r = / sin"* </.r, 
Jo 



and that 



i7r ^2,»„ ^_ x -3-5 • • • {2m- i)7C 



P 



sin 2 '"* dx = 



2. 4.6 ... 2?» 2 

sin 2 «+i^ <£c = 



/ sin 2 «+i^- dx = * 

^0 3-5-7 . . . (2»*+ i)' 



when f» is a positive integer. 



28. f { ——U-dx = 6. Put *z = i. 



1 (x - xrf 

X* 

29. f M (x ~ 2) l dX - = 8 + f l/J-^r. Put x - 2 = * 

30. r^ W"-* rf* = 4 - jr. Put ,* - 1 = A 

Jo <->* + 3 

01 / — 1 — , „ = — ■ . Where a > b. 

31. / fl 4- b cos Va 2 _ p 

32 f x s ^ n x dx = *• 

Jo 

150. A Theorem of Mean Value. — Since in 

t/.r 

<£c keeps the same sign throughout the summation, 

XX X 

mf dx < { f{x) dx < M f dx, 

where m and J/ are the least and greatest values respectively of the func- 
tion f{x) in {x , X). Therefore the integral lies in value between 
m(X — x ) and M(X — x Q ). 

Sincey^Jt) is continuous in the interval, there must be a value of or, 
say £, in (x Q , X), for which 

( X f(x)J X = (X-x )f{$), 

f{£) being a value of the function between m and M, its least and 
greatest values. 



Art. 150. J ON DEFINITE INTEGRATION. 

The value 



219 



<& — a j Xo 



is called the mean value of the function in (a , A"). 
If F{x) is a primitive off[x), then 

F{x)-n-\) = (x--\)A$), 

since F'(x) =/(x). This is the familiar form of the Law of the 
Mean as established in the Differential Calculus. 

The theorem of mean value for the Integral Calculus can be estab- 
lished directly from the definition of a mean value. For, if 

Ax == (X - x )/n, 
then 

f/[x)dx=£ 2 Jxf(x r ), 

= '(X-x ) f A*o)+A*\) + ..-+A**) m 

« =00 

If the limit of the arithmetical mean of the n values of the function 
at the points of equal division of (x , X) be indicated by /([£), the 
result is the same as above indicated. 



If y=/(x) 

^y dx ■ 
x 






Geometrical Illustration. 
is represented by the curve AB, then 

: area (x Q AZBX). 



This area lies between the rectangles 
x ATX and x SBX, constructed with x Q X 
as base and the least and greatest ordinates 
to the curve respectively as altitudes. 
There is evidently a point c, between x Q 
and X at which the ordinate £Z = /(q) 
is the altitude of a rectangle x Q RQX, inter- 
mediate in area between the greatest and 
least rectangles, whose area is equal to that 
bounded by the curve. 




EXAMPLES. 

1. Find the mean value of the ordinate of a semi-circle, supposing the ordinates 
taken at equidistant intervals along the diameter. 



Let x 2 -f- y 2 = a 2 be the circle. Then 



1 /.+« _ 
2a.f_„ r 



x 2 dx = \rta, 



viz., the length of an arc of 45' 



220 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XIX. 

2. In the same case, suppose the ordinates drawn through equidistant points 
measured along the circumference. Then the arc length is the variable, and the 
mean ordinate is 



2 

sin dQ = —a. 
7tJ n it 



7tJ 



We shall see later that this is the ordinate of the centroid of the semi- 
circumference. 

3. A number n is divided at random into two parts; find the mean value of 
their product. 



n Jo % 



x)dx = — n 2 



4. Find the mean value of cos x between — n and -J- tt- 

5. If Af r *(y) is the mean value of y = f{x) in (x v x 2 ), show that: 

(a). M*{2x*+zx- i) =8i. 

(b). M]{2 _3*+5*i-*i)=Y. 

(c). M*(x+ i)(* + a) = i2i. 

(d). M Q in (sin B) — 2/tt. 

6. Find the mean distance of the points on the semi-circumference of a circle of 
radius r, from one end of the semi-circumference, with respect to the angle. 



Wo 



M^-- 2r cos dO = ¥-; 

Tt 



By the mean value of n numbers is meant the »th part of their 
sum. To estimate the mean value of a continuous variable between 
assigned values, we take the mean of n values corresponding to equi- 
distant values of some independent variable and find the limit of this 
average when the number of values is increased indefinitely. The 
mean value depends on the variable selected. See Exs. i and 2 above. 

liy is a function of /, then the mean value of y with respect to / 
for the interval {f lt Q is 



1 /»'a 



dt. 



151. An Extension of the Law of the Mean. — If <p(x) and ip(x) 
are two continuous functions of x, one of which, ip(x), has the same 
sign for all values of x in (.v , X), then we shall have 

f X <P(x)lf,(x)dX =: cP(£)f X l/;(x)dx, 

JXq J Xq 

where £ is some number between x Q and X. 

For if m and M are the least and greatest values of <fi(x) in 
(x , X), then the integral must lie between the numbers 

mi ip(x)dx and Ml t/:(x)dx, 

since tp(x) dx does not change sign in (.v , X). Therefore there 



Art. 152.] ON DEFINITE INTEGRATION. 221 

must be a number £ in (.v , X) for which the integral has the value 
proposed, since 0(-v) is a continuous function. 

152. The Taylor-Lagrange Law of Mean Value. — Integration 
by parts furnishes a simple and an elegant method of deducing the 
important formula of Lagrange, and gives the form of the remainder 
in a much more useful form than that of the Differential Calculus. 

Let z be a variable in the fixed interval (a, x). Then 

/(•*) -a*) = £/'« * = - £7» 4* - «). 

Put u = f'(z) f dv = d{x — z), and integrate by parts. 

••• /(-*•) -a*) = - (* - «)/"(*)iC + jf (* - »r(«)*. 

= (* - ")/'(") - f'(* - •)/"(*) 4* - 2)- 

Put « =y r// (2), </» = (.v — z)d{x — z), and integrate the 
integral on the right by parts. Then 

/(*) -A«) = (*-«l/ , W+^=j f ^/"W- fy^f"(»)Hx—). 

Continue to integrate by parts in the same way, and there results 

n 
r=o 

This is Lagrange's theorem with the terminal term expressed as 
a definite integral. This form of the terminal term shows that the 
difference between the function f{x) and the series vanishes when 
n = 00 , provided 

\x - ,)- 



£■ 



-/»"(*) = o (2) 



for all values of z in (a, x); and moreover, if this limit is not zero 
for any finite subinterval of (a, x), however small, the terminal 
term does not vanish and the series, although convergent, cannot be 
equal to the function.* 

The law of the mean expressed in § 151 enables us to transform 
the definite integral in (1) directly into the forms of the terminal 

*The reader should be warned against the language of many writers who con- 
found the remainder of Taylor's series with the terminal term of the law of the 
mean, for they may be quite different. In fact, if Taylor's series 6"oo is convergent 
and S M = S n -\- R n , then we should write 

f(x) = S n + Rn+ T n . 

The terminal term being R n -\- T n . In order that f(x) = S^ it is necessary that 
both^^„ — o, £T n = o. £R n — o does not ensure £ T n — o. See Appendix. 
Note 8. 



222 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XIX. 

term given in the differential calculus. For, since {x — z) p keeps 
its sign unchanged for all values of z in (a, x), we have 



/ 



(* - *)" ~H,.l * _ (* - S ) 



n . 



r+\z) dz = {x f) _ /" +i (g) f\ x - *)' *' (3) 



where ^ is some number between a and x. This result, (3), takes 
Lagrange's form when p — ?i, and Cauchy's when p = o. The more 
general form (3), where /> is any integer, is due to Schlomilch and 
Roche. 

153, The Definite Integral Calculated by Series.— If f{z) can 
be expressed in powers of (z — a) by Taylor's series, for all values 
of z in (a, x), then also can the primitive of f{z), and the definite 
integral of the function is equal to that of the series, taken term by 
term, between a and x. Hence, integrating between a and x, 

A*) =A") + ( 2 - ")/'(") + ^r^-V"(«) + • • • , 

we have 

£/{z) dz = (x-a)/{a) + £=#/"(«) + ^=^/"(«) + • • . (■) 

In particular, put at = o, then we have 

£/(z) dz = a -f(a) - a l/'(a) + a l/"(a) -..., (,) 

a formula due to Bernoulli. 

Knowledge of the derivatives at a serve therefore to compute the 
integral. When a = o in (1), then 

pW dz = a/(o) + £/'(o) + ~f\o) + ..., (3) 

which is Maclaurin's form, and is more convenient, in general, for 
computation than (2). 

EXAMPLES. 

1. Deduce Bernoulli's formula (2), § 153, by using the formula for parts, 
ffix) dx = xf(x) -fx/'(x) dx. 

J-x \ 3 5 2! 7 3! / 



3. / *-* a dx — x ■ - -f . .. 

Jo 3 5 2! 7 3! 

4. ^log (tan <p)d<p = - ^_i. + ij_l + ...J. 

154. Observations on Definite Integration. — In order that a 
function may admit of definite integration in an interval {a, ft) it 
must, in general, be one-valued and continuous throughout the 



Art. 154.] ON DEFINITE INTEGRATION. 223 

interval. If the function is not one-valued, then generally the 
branches must be separated so that each may be taken as a one- 
valued function. If the function becomes infinite for any value of 
the variable between the limits of integration, then for such particu- 
lar values of the variable the integral must receive special investiga- 
tion, a case which we do not consider in this text. 

In definite integration when one of the limits is infinite, we 
consider the integral 



I 



f(x) dx 

as the limit to which converges the integral 

X /{x) dx, 



£ 



when x = 00 , provided there be such a limit. The same remark 
holds when one limit is — 00 and the other -(- 00 . 

All continuous one-valued functions are integrable in the interval 
of continuity, as demonstrated in the Appendix, Note 9. But all 
continuous one-valued functions are not differentiable (see Appendix. 
Note 1). 

The study of definite integrals will be taken up again in Book II. 



EXERCISES. 

dx ,— _ C a dx 

_=:2^ 2. / — 

ya — x t/o 4/, 



ax — x" 



Jfi dx /** 

I ' . = \%. 4. / sin-J.r dx — \tt — I. 

j x \/x 2 — I «/o 

5. r dX =-*-. 6. /*Wr dx = log 2* - J. 

J e 2 -f COS X 3 4/3 J 

"t/o I + cos cos x ~ sin ' J 1 
pi* dx it 



' 1 -(- cos 9 cos x sin 



10. 



o a 1 sin 2 * -\- b* cos 2 * lab 

/•*«■ dx 7i(a 2 -f- P) 

J (a 2 sin 2 * + b 2 cos 2 *) 2 ~~ ^P ' 



/»/3 dx <t% fl* sin x dx , M 1 

11. / — = it. 12. / — ; r- = ±tt 4- tan-* — _ 

J a \/(x - a)(/3 - x) Jo l + cos 2 * ^2 

13. Show that, when & < 1, 



Jo */i-£ 2 sin 2 9 2 L W V 2 ^/ \2-4-6/ T J 

ptic integral. 

/ \ W 2 _l_ 1 ^ »2 + 2 2 1- -r 2n ,j 4 



This is an elliptic integral. 
14. Show that 



224 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XIX. 

Put dx = i/n. The limit of the sum is then 

J I* 1 dx _ it 
o l + x ' 1 ~ 4~" 



15. Show that the limit of the sum 



+ -J = + ...+ 



|/» 2 - I 2 |/»2 - 2 2 |/« 2 - (»— I) 2 

when « = oo , is ^7T. 
16. Show that 

J' sin mx sin nx dx and / cos mx cos «jc dx 
t/0 

are zero when m and « are unequal integers, and are equal to \n if m and n are 

equal integers. 

sin 2 x cos 3 jf dx = T 2 ^. 

«« /*i 7r sin 4- cos „ , , _ , . . - 

18. / -^ — 7- </0 = i log 3. Put sin — cos 6 = x 

Jo 3 -f sin 2& 5 ° 

19. / ^ — = log 3. Put # — *-* = z. 

J [ x |/x* -(- 7-r 2 -f- I 

20. /•" *^ = _*_,. 21. p«.dSr=i. 

Jo I — 2a cos x -f ^ I — a- 1 ,/ 

24 - i ?+^ = ss • 25 - i ^r+^F = * + log *• 

oc f° x dx n _ /•«» dx it 

28. /" * = *. 29. f ,- £A , - ^ 

J- 1 |/2 + X — X 2 Jo ( X + a ~l~ 



la 



tan x dx — log |/2. 31. / sec 2 .r dx = 1. 

«/o 

32 ' J a 2 + ^ 2 -T^c3os"e =: ^r72- 33 - J o -y| _ ^r~^ = 

34. f x\i - xfdx = 2 f' n sin°Q cos*0 d( 
Jo Jo 



1 1 5" 

/0 



35. fx 2 (i - x 2 )Vx = /"-'"sin 2 © cos»0 dr 

Jo Jo 



32 



36. p(i -,)•* = £. 

37. Putting <«* _ 1 = y\ show that 
.logs ,1 ;/2 dy 4 _ ^ 



/ |/V* - I dx = 2 J -±— *-= = 



Art. 154.] ON DEFINITE INTEGRATION. 225 

38. If x + 1 = V, 

f \/zx + x* dx = £ Vj' 2 - * d >' = VJ- * Io g( 2 + VD< 

39. Tutting x = a sin 6, 

I x 1 \/a z — x 2 dx = a* I sin 2 G cos-0 </Q = — ^ 

40. If * = a tan G, 

-a x*dx /»*„ sin 3 _ 3 \/z~ 4 

Jo (^ + 0» == V.. c^Q^- 



a. 

2 



PART IV. 

APPLICATIONS OF INTEGRATION. 
CHAPTER XX. 

ON THE AREAS OF PLANE CURVES. 

155. Areas of Curves. Rectangular Coordinates. — The sim- 
plest method of considering the area of a curve is to suppose it 
referred to rectangular coordinates. The area bounded by the 
curve, the jvr-axis, and two ordinates corresponding to the values 
x , x x of x, is represented by the definite integral 



J 1 y dx. 
x 



This has been shown to be true in Chapter XVI, as an illustra- 
tion of the definite integral. It has been shown that the definite 
integral is independent of the manner in which the ordinates are dis- 
tributed in making the summation. 

We demonstrate again that the definite integral gives the area in 
question. For simplicity we divide the interval (x , xj into n 

equal parts, each equal 
to Ax. Let AB be the 
curve representing the 
equation y =f(x), and 
x Q ABx x the boundary of 
the area required. Let 
MN be one of the sub- 
divisions of x Q x v Draw 
ordinates to the curve at 
each of the points of 
■x division, and construct 
the ?i rectangles such as 
FlG - 6 9- MPgN, and also the n 

rectangles such as MpQN. Since the curve is continuous, we can 
always take Ax or ylAVso small that for each corresponding pair of 
rectangles the curve PQ lies inside the rectangle PpQq, and therefore 
the area MPQN of the curve lies between the areas of the rectangles 
Hence the whole area x Q ABx l for the curve 

226 




MPqN and MpQN 



Art. 155.] 



ON TIIK AREAS OF PLANK CURVES. 



227 



lies between the sum of the rectangles represented by MPgNand the 
sum of those represented by MpQX. The difference between the 
sums of these rectangles is the sum of n rectangles of type PpQq. 
Which sum is equal to a rectangle represented by BR, whose base 
BS is Ax and altitude RS is y x — y , where y x = x x B, y = x A. 
( \\, r ) being the greatest and least ordinates in the interval. When 
the number of rectangles, n, is increased indefinitely, the difference 
between the sums of the rectangles, the one greater, the other less, 
than the curved area, converges to zero. Therefore the sum of 
either set of rectangles has for its limit, when n = 00 , the area of 
the curve, or 



£ 2yAx = / ydx. 

M=ao I Jx n 



If y = f(x) is the equation of a curve, the area A included 
between the curve, the ordinates y ,y 1 at x Q , x lt and the x-axis is 



i>> 



dx. 



EXAMPLES. 

1. Area of the circle. 

Taking x 2 -f- )' 2 = d 2 as the equation of the circle, 



.-. y — 


± \/a 2 — x 2 . 




y 


-, 


3 . 


P 










Pi 




/ 











a 


'0 £ 


c a 


\A 



Fig. 70. 
Taking the positive value of the radical, we have for the area x P P x x x , 

If x x = a, we get the area of the semi-segment x P A. If x = o, and x x = a, 
we have the area of the quadrant OB A equal to 



i> 



x' 2 dx = \ita 2 . 



If is the angle POA, then y = a sin 0, x — a cos 0. 

. •. </j= -asinfi dQ. The area, A, of the circular quadrant is then given by 

A = [ a y dx = - a? f sin 2 </9 = a 2 f^sirMdQ, 

= \a\B - sin cos fl)]-"" = \na\ 
The area of the entire circle is therefore ltd 1 . 



22; 



APPLICATIONS OF INTEGRATION. 



[Ch. XX. 



2. The area of the ellipse. 

From the equation of the ellipse x 2 /a 2 -f- y 2 /b 2 = i, we get/ = — \/d l — 

Consequently, as in Ex. I, the area of the elliptic quadrant is 

b 



If 






M 








P 







N 


M 




P' 







a t/0 



which is b/a times the corresponding area of a circle 
of radius a. Hence the area of the entire ellipse is 
Ttab. 

3. Area of the parabola. 

Taking y 2 = px as the equation of the curve, 
and the positive value of the radical in y = \/px, we 
have the curve OP. The area OPN is then 



Fig. 71. 



f* VP* <** = $P i **X> 



- %xy. 

But xy is the area of the rectangle ONPM. The area of the segment POP' of 
the parabola cut off by a chord perpendicular to the 
diameter is two thirds the rectangle MPPM'. 

4. Area of the hyperbola. 

Let x 2 /a 2 — y 2 /b 2 = 1 be the equation to the 
curve. Then the area of A PN is 



A = J y dx, 



a 2 dx. 



log (x + \/x' 



y 








P/ 


\^Y 








i>^~. 







A 


T 




= \xy — \ab log 



5. Area of the catenary. 
The equation to the curve is 



y =\a\e« + e « ) . 
. O VPN is 

x C x l~ -~\ 

A = J \a \e a + e a ) dx, 



The area O VPN is 



Fig. 73. 



= U 2 \e a — e a ) — a 



Vy' 1 



If NL is perpendicular to the tangent at P, show that the above area is twice 
that of the triangle PLN Observe that tan LNP = Dy, LN = y cos LNP, etc. 

6. Show that the area of a sector of the equilateral hyperbola x 2 — y 2 = a 2 
included between the .r-axis and a diameter through the point x, y of the curve is 

7. Find the entire area between the witch of Apiesi and its asymptote. 

The equation is (x 2 -\- 4a 2 ]y = 8tf 3 . Ans. \-itd 1 . 



Art. 155.] 



ON THE AREAS OF PLANE CURVES. 



229 



8. Find the area between the curve y = log x and the .v-axis, bounded by the 
ordinates at x = I and x. Ans. x(log x — I) -f- I. 

9. Find the area bounded by the coordinate axes and the parabola x^ -\- y^ = at. 

Ans. \a 2 . 

Ans. \nab. 



10. Find the entire area within the curve 



©'+ ©*- 



11. Find the entire area within the hypocycloidx\ + 7* = a--. 
Hint. Put x = a sin :< 0, y = a cos :t 0. 



Ans. |7Ttf 2 . 



12. Find the entire area between the cissoid (2a — x)y 2 = X s , and its asymp- 



tote x 



Ans. 



13. Find the area included between the parabola x 2 = 4.av and the witch 
y(x* + 4* 2 ) = 8a 3 . Ans. a\2it - f ). 

The origin and the point of intersection of the curve give the limits of the 
integral. 



14. Find the area of the loop of the curve 

/2 — (r _ n\lr _ h\1 



Hint. Let x 



z 2 . Ans. 



1 \V- a ? "Q 

*5\ < 




15. Find the whole area of the curve a 2 y 2 — x 3 (2a — x). 



Fig. 74. 



Ans. 




Fig. 75. 



16. Find the area of the loop c_ Ihe curve 

ay - x\b -f x). 
The area of the loop is 

. 2 /»° 32/^ 

A = S I X 2 A/b + X dx - ■ 5 

W_* T ^ 3-5-7^ 

Put 3 -f x = z 2 . 



17. Show that iiy =f(x) is the equation of a curve referred to oblique coordi- 
nate axes inclined at an angle go, then the area bounded by the curve, the x-axis, 
and two ordinates at x , x x is 

A = sin go I y dx. 

18. The equation to a parabola referred to a tangent and the diameter through 
the point of contact isy 2 = kx. 

Show that the area cut off by any chord parallel to the tangent is equal to two 
thirds the area of the parallelogram whose sides are the chord, tangent, and lines 
through the ends of the arc parallel to the diameter. 

19. The equation to the hyperbola referred to its asymptotes as coordinate axes 
is xy = c 2 . If (» is the angle between the asymptotes, show that the area between 
the curve, x-axis, and two ordinates at x , x l is 

c 2 sin go log ( — ) • 

20. If y = ax n is the equation to a curve in rectangular coordinates, show that 
the area from x = o to x is 



2 30 



APPLICATIONS OF INTEGRATION. 



[Ch. XX. 



156. If the area bounded by a curve, the axis of y s and two abscis- 
sae x , x v corresponding to the ordinatesj' , y v is required, then that 
area is 

pyx 
A = J x dy. 



EXAMPLES. 

1. Find the area of the curve/ 2 = px between the curve and the j-axis from 
y ■= o to y = y. 

2. Find the area of the curve y = e x between the curve, the j-axis, and ab- 
scissae at y = 1, y = a. Check the result by finding the area between the curve 
and the jf-axis for corresponding limits. 

Also find the area bounded by the curve, the /-axis, and the negative part of. the 
^r-axis. 

157. Observe that in the examples thus far given the portion of 
the curve whose area was required has been such that the curve was 

wholly on one side of the 
axis of coordinates. 

It is evident that if 
the curve crosses the axis 
# between the limits of in- 
tegration, then, y being 
positive above the ^f-axis 
FlG ' 76, and negative below it, 

those portions of the area above Ox are positive, those below are 
negative. The integral 

~ ri ydx 




£ 



is then the algebraic sum of these areas, or the difference of the area 
on one side of Ox from that on the other side. 



EXAMPLE. 

Find the area oiy = sin x from x = O to x 
We have 



Jo 

f 

I! 



'T- 

Jo 



: I ' sin x dx = — cos x 
h _io 

But / sin x dx = 2, 

'2rr 




sin x dx = — I. 



Fig. 77. 



... A*" = A" + A*" = 2 



158. It is evident that the area considered can be regarded as the 
area generated or swept over by the ordinate moving parallel to a fixed 
direction, Or. 




Art. 160.J UN THE AREAS OF PLANE CURVES. 

If we have to find the area between two 
curves 2/ 

y x = 0(.v), r 2 = ip(x), 

and two ordinates at a and b t such as the 

area LMXR in the figure, that area can be 

computed by finding the area of each curve 

separately. But if it is more convenient, the (J 

area is Fig. 78. 

j\y, ~J\) dx = j\,/:(x) - <p{x)1 dx. 

The area in question is generated by the line P l P 2 , equal to the 
difference of the ordinates y % — y v moving parallel to Oy from the 
position RL to NM. 

EXAMPLE. 

Find the area bounded by the curves 

x(y — e x ) — sin x and 2xy = 2 sin x -\- x*, 
the j-axis, and the ordinate at x = I. 

A = j {e* - \x 2 ) dx = e-%=zi.$S +. 

It would not be so easy to find the areas of each curve separately. 

159. If it be required to find the 
whole area of a closed curve, such as that 
represented in the figure, we may proceed 
Jr^ as follows : 

Suppose the ordinate MP to meet the 

curve again in Q, and let MP =J> V 

-x^IQ = y r Let a and b be the abscissae 

of the extreme tangents aA and bB. 

Then the area of the curve is 

A = jf (> ' 2 "^ dX% 
This result also holds if the curve cuts the axis of x. 
EXAMPLE. 

Find the whole area of the curve (y — tnxf - 
Here 




y = mx ± \/a 2 

yx 

A 



mx 
mx 






(y 2 -}'i) dx > 



x 2 dx = na 2 . 

160. The area of any portion of the 
curve 



K'T 



(') 




Fig. 80. 



232 APPLICATIONS OF INTEGRATION. [Ch. XX. 

is ab times the area of the corresponding portion of the curve 

f{ X , y) = c. (2) 

For (1) is transformed into (2) by puttings = ax' ', y = by' in (1); 
and hence y dx, from (1), becomes ab y' dx' , and we have 



Cy **=*£'*• 



EXAMPLES. 

1. The entire area of the circle x 2 -j- y 2 = 1 is n. Hence that of the ellipse 
x 2 /a 2 + y 2 /b 2 = I is abit. 



2. Find the whole area of the curve ( — J -|- l^-\ = 
In Ex. 11, § 155, it is shown that the area of 



X i _J_ y\ _ J 

is f 71. Hence that of the proposed curve is \itab. 

3. Check the result in Ex. 2 by putting x = a sin 3 0, y — b cos'0. 
Then ydx = ^ab sin 2 cos 4 d(p. 

I sin* 2 cos 4 d(p = %rtab. 


161. Sometimes the quadrature of a curve is to be obtained when 
the coordinates are given in terms of a third variable, or is facilitated 
by expressing the coordinates in terms of a third variable. Thus if 

x = 0(/), y = tit), 
the element of area is 

ydx— ip(t)cp'{t)dt. 

EXAMPLES. 

1. Find the area of the loop of the folium of Descartes, 

x z ._j_yj _ ^axy. 
Put y — tx\ then 

- _3fL_ v - 2afi 

I - 2 ' 3 

•'• ^=or+^p 3 ^ and 




fydx = 9 a 2 f-^- i 



/ 2 (i - 2/- 3 )<# 6a 2 ga 2 



Fig. 81. y ~* J (i + z 3 ) 3 1 4- *» 2(1 + *»)*• 

The limits for / are o and 00 . Hence A = |a 2 . 
2. In the cycloid, 

x — a{t — sin /), y = a(i — cos /), 

. •. f y dx = a 2 /Vers 2 / dt = q& J sin* \t dt. 

Taking t between o and 7T, we get T,na 2 for the entire area between one arch of 
the cycloid and its base. 



Art. 162. J ON THE AREAS OF PLANE CURVES. 233 

3. Eind the area of the ellipse using x' 2 /a' 2 -\- y' 2 /b' 2 = 1, where x = a cos <p, 

y ■=. b sin 0. 

4. Find the area of the hyperbola x 2 /a' 2 — y 2 /b 2 = 1, from x = a to x = x, 
using x = a sec <p, y = b tan 0. 

162. Areas in Polar Coordinates. — Let p —/(0) be the polar 
equation to a curve. We require the area of a sector, bounded by 
the curve and two positions of the radius vector, corresponding to 
6 = a, = ft. 




Fig. 82. 

Let AB represent p =f(6), 01 the initial line. /_IOA = a 9 
/_IOB = /3. Then OAB is the sector whose area is required. 
Divide the angle A OB = ft — a into n equal parts each equal to 
AS, and draw the corresponding radii cutting AB in corresponding 
points P, Q, etc. ; dividing the curve AB into n parts, such as PQ. 
Through each of the points of division draw circular arcs with center 
0, such as Qp, qP, etc. From the continuity of p =f(6), we can 
always take Ad so small that the sector OPQ of the curve lies 
between the corresponding circular sectors OPq and OpQ, and there- 
fore the area of the whole sector OAB lies between the sum of the 
circular sectors of type OPq and the sum of the circular sectors of 
type OpQ. But the difference between these sums of circular sectors 
is equal to the area 

ALNM= \{0B* - OA 2 )A6, 

which has the limit o when A6( = )o, or when n = 00 . Therefore 
the sum of either the external or internal circular sectors converges 
to the area of the sector OAB as a limit when n = 00 . 

Putting p Q = OA, p n = OB, and p r {r = 1, 2, . . .), for the 
radii to the points of division of AB y the area of the curvilinear 
sector OAB is 



234 APPLICATIONS OF INTEGRATION. [Cii. XX. 

EXAMPLES. 

1. Find the area swept out by the radius vector of the spiral of Archimedes, 
p = ad, in one revolution. 

We have A = £ f p 2 JO = \ f aW JO = §jr 3 a 2 . 

Jo ' Jo 

2. Find the area described by the radius vector of the logarithmic spiral 
ft — e a& , from = o to = \it. Arts. — (e na — i). 

3. Show that the area of the circle p = a sin is £7tvz 2 . 

4. Find the area of one loop of ft = a sin 20. Ans. \ita 2 . 

5. Find the entire area of the cardioid ft = a(i — cos 0). Ans. \ita 2 . 

6. The area of the parabola p = a sec' 2 \0 from = o to — -^tt is |# 2 . 

7. Show that the area of the lemniscate p 2 = a 2 cos 20, is a 2 . 

8. In the hyperbolic spiral ftO = a, show that the area bounded by any two 
radii vectores is proportional to the difference of their lengths. 

9. Find the area of a loop of the curve ft 2 = a 2 cos nO. Ans. a 2 /n. 

10. Find the area of the loop of the folium of Descartes, 

x 3 + y* = saxy. 
Transform to polar coordinates. Then 

3<z cos sin 



ft = 



sin 3 -j- cos 3 



Therefore the area is 

u 2 du 



9a' /•— sin*0 cos z JO _ 90* /» w 

~2~ J (sin*0 + cos 3 ©) 2 - -J J ( 7 



4- « 3 ) 2 2 
where u = tan 0. 

11. Show that the whole area between the curve in Ex. 10 and its asymptote is 
equal to the area of the loop. 

12. Find the area between the curves 

*■+*■= (~y and p 2 +o 2 = (~y. 

13. The area of ft = a cos 30, from o to ^7f, is faita*, 

14. Show that the area of ft = a (sin 20 -f- cos 26), from o to 271", is 7nz 2 . 

15. The area of p cos = rf cos 20, from o to \it, is £(2 — \it)a 2 . 

163. We come now to consider the area generated by a straight- 
line segment which moves in a plane, under certain general conditions. 
In rectangular coordinates we have considered the area generated by 
the moving ordinate to a curve. In polar coordinates the area con- 
sidered was generated by a moving radius vector. In the former case 
the generating line moves parallel to a fixed direction, in the latter it 
passes through a fixed point. 

A point Q is taken on the tangent at P to a given curve PP' , such 
that PQ = /. To find the area bounded by the given curve, the curve 
QQ' described by Q, and two positions PQ, P'Q' of the generating 
line. 



Art. 16: 



ON THE AREAS OF PLANE CURVES. 



35 



Let PQ = /, P'Q' = t+Jt, PI= 6/, P'/ = S't, and 6 be the 
angle which the tangent at P makes with a fixed direction. Let A A 
represent the area swept over by PQ in moving from PQ to P'Q' through 



Q^X' 1 ' 




Fig. 83. 

the angle Ad. Draw the chord PP' and the circular arcs QM, Q'M' 
with /as a center. Then A A is equal to the area of the circular 
sector QIM, plus a fraction of the area of the triangle PIP' , plus a 
fraction of the area QM Q'M' . Or, in symbols, 

A A = £(/ - o7)M# 



/, 



/, 



_|_ ^_L<y/. tf V sin J#-f^[(/-f- J/ + S'tf - (/ - tf/) 2 ]J#, 

where y^ , _/* 2 are proper fractions. Observing that z?/, 67, and dV 
converge to o when A6( = )o, divide by AS and let A6(=)o. Then 

dA 

or ^4 

Hence between the limits 6 =. a, 6 = /? the area swept over by 



i/ 2 ^. 



/IS 



=*r 



> 2 ^. 



When the law of change of /, the length of the tangent, is given 
as a function of 6, the area can be evaluated. If t — f(6) be this 
relation, the curve t =_/"(#), considering / as a radius vector and # 
the vectorial angle, is called the directing or director curve of the 
generating line. 

EXAMPLES. 

1. Show that the area swept over by a line of constant length a laid off on the 
tangent from the point of contact is 7ta' 2 , when the point of contact moves entirely 
around the boundary of a closed plane curve. 

2. The tractrix is a curve whose tangent -length is constant. Find the entire 
area bounded by the curve. (Fig. 84.) 

The area in the first quadrant is generated by the constant length PT = a 
turning through the angle \it as the point P moves from J along the curve J PS 
asymptotic to Ox. Therefore the area in the first quadrant is Irta 2 , and the whole 
area bounded by the four infinite branches is ita 2 . 



236 



APPLICATIONS OF INTEGRATION. 



[Ch. XX. 



3. Check the above result by Cartesian coordinates and find the equation to the 
tractrix. 

We have directly from the figure 

dy 



—-— - tan PTN 

dx 



yd* 



: . . ' . y dx = — ya 2 — y 2 dy. 




Fig. 84. 



Hence the element of area of the tractrix is the same as that of a circle of 
radius a. It follows directly that the whole area of the tractrix is rta 2 . This 
gives an example of finding the area of a curve without knowing its equation. To 
find the equation of the tractrix, we have 



dx 



\/a 2 



dy 



Integrating, we get 



x = — ya 2 — y 2 -j- a log 



\/a 2 - y 2 



since x = o wheny = a. This is said to be the first curve whose area was found 
by integration. 

4. Show that the area bounded by a curve, its e volute, and two normals to the 
curve is 



*o 



where p is the radius of curvature of the curve, and Q the angle which the normal 
makes with a fixed direction. 

164. Elliotts Theorem. — Two points P x and P on a straight 
line describe closed curves of areas (P^ and (P 2 ). The segment 
P X P 2 moves in such a manner as to be always parallel and equal 
to the radius vector of a known curve p =/"(#) called the director 
curve. 

It is required to find the area of the closed curve described by a 
point P on the line P X P 2 which divides the segment P X P 2 in constant 
ratio. 



Art. 164.] 



ON THE AREAS OF PLANE CURVES. 



237 



Let (P), (P x ), (P 2 ), (A) be the areas of the closed curves 
described by the corresponding points as shown in the figure. Let 





Fig. 85. 

P X P 2 and P X P 2 , Fig. 86, be two positions of the segment. Produce 
them to meet in C. 



.--^ 




Let p = P X P 2 , P,P/PP 2 = mjm 2 . 



PJ> = 

m l -\-m 2 



*—P x P t = kj>, 



PP. 



^p x p 2 = k 2 p, 



(1) 



2 m l -f m 2 
where k x -f- k % = 1 . 

The element of area P X P 2 P 2 'P X is, § 163, if CP X — r, 
d(P 2 ) - d(P x ) = |(p + rf dd - JH dd, 
= pr dd + \tf dd. 
In like manner the element of area P X PP'P X ' is 

d(P) - d(P x ) = l(k lP + rf dd - p dd, 

= k x pr de + %k x Y de. 

Multiply (1) by k x and eliminate k x pr dd between (1) and (2), 
remembering that k x -f- k 2 = 1. Then 

d(P) = WJ + w\) - *AHA)- 

Integrating for a complete circuit of the points P x and P 2 about 
the boundaries of the curves, we have 

{P) = K(Pd + K( p .) r KK( A ), (3) 

where the area of the director curve is given by 

(A)=ffffid0, 

the limits of the integral being determined by the angle through 
which the line has turned. 



238 APPLICATIONS OF INTEGRATION. [Cn. XX. 

In particular, if P X P 2 = p is constant and equal to a, we have 
Ho/ditch's theorem, 

{P) = kjft + K{P 2 ) - RV/' /ft 

If a chord of constant length a moves with its ends on a closed 
curve of area (C), the area of the closed curve traced by the point 
P which divides the chord in constant ratio m : n is 

(P) = (C) 



(in -f n y ' 

= ( C ) ~ c x c 2 7t, 
if P is distant c and c 2 from the ends of the chord. 

EXAMPLES. 

1. A straight line of constant length moves with its ends on two fixed intersect- 
ing straight lines; show that the area of the ellipse described by a point on the line 
at distances a and b from its ends is nab. 

2. A chord of constant length c moves about within a parabola, and tangents 
are drawn at the ends of the chord; find the total area between the parabola and 
the locus of the intersection of the tangents. Am. \itc l . 

The area between the parabola and the curve described by the middle point of 
the chord is the same. 

3. It can be shown that the locus of the intersection of the tangents in Ex. 2 
to the parabola y 2 = \ax is 

{y 2 - 4ax){y* -f 4rt' 2 ) = a 2 c 2 . 

Check the result in Ex. 2 by the direct integration 



/ 



z dy = \c 2 it 



fromjj' = - 00 to/ = -f 00. z being half the distance from the intersection of 
the tangents to the mid-point of the chord. 

4. Tangents to a closed oval curve intersect at right angles in a point P\ show 
that the whole area between the locus of P and the given curve is equal to half 
the area of the curve formed by drawing through a fixed point a radius vector 
parallel to either tangent and equal to the chord of contact. 

5. If p v B x and p 2 , B 2 are the polar coordinates of points P x and P 2 on a straight 
line, then the radius vector p of a point on this straight line which divides the 
segment P X P 2 — X so that PP X = k x X, PP 2 = k 2 \, is determined by 

P 2 = k -iP\ + V2 2 - k A Xi ' (0 

This is Stewart's theorem in elementary geometry. If (p is the angle which p 
makes with P 1 P 2 , then 

Px > = p* + k*W - 2k x Xp cos 4>, 
p 2 2 = p 2 + k*X* + ik 2 \p cos 4>. 

The elimination of cos <p gives (1) at once. 
Multiply (1) through by idfj, then 

|p 2 dQ = k 2 Ip* JO + k x ip^ dO - k x k % \ X 2 ttB, (2) 

or d{P) = k 2 d{P 2 ) + k x d(P x ) - V' 2 "M> 

and Elliott's theorem follows immediately on integration. 



= -K> 


+ !*? & - \ 


r 2 dQ, 


= ;-/, 


dO -f U? dO. 




= lr' 


dO - \{r - /, 


Y do, 


= r/ 1 


dO - \l* dO. 




= /. 


r dO + m - 


l*)d 



Art. 1O4.J OX Till-; AREAS OF PLANE CURVES. 239 

The geometrical interpretation of (2) is as follows : Let A. = P x l\ be constant 
Construct the instantaneous center of rotation / of X as 1\1\, turns through JO. 
Then /',/','. PP\ J'JV (Fig. SO) subtend the angle M at /. The center / being 
considered as origin or pole. (2) follows at once. The extension to the case when 
A. is variable is immediately evident. 

6. Theory of the Polar Planimeter. 

In Fig. 86, let P X P 2 = / be constant. At P let there be a graduated wheel 
attached to the bar P X P 2 in such a manner that the ;ixk- of the wheel is rigidly 
parallel to P x P r This wheel can record only the distance passed over by the bur at 
right angles to the bar. 

Let P 1 P=l 1 , PP 2 -l v Let CP = r. 

Then with the symbolism of § 164 we have 

d(P 2 ) - d(P } 
d(P) - d(P,) 

Adding these two equations, 

d(P 2 ) - d{P,) 

But r dO = dR is the wheel record for a shift of the bar. 

Integrating, we have for the area bounded by the curves traced by P 2 and P x 
and the initial and terminal position of the bar 

W - (*\) = K R 2 - *i) + Wi - A 2 )(9 2 - 9i), 

r 2 being the initial and terminal angles which the bar makes with a fixed 
direction, and P l} P 2 the initial and terminal records of the wheel. 
Notice that when the wheel is attached to the middle of the bar 

(P 2 ) - (/\) = 1{R 2 - P x ). 

The path of P x is a circle in Amsler's instrument. 



EXERCISES. 

1. Find the area of the limacon p = a cos -\- b, when b > a. 

Ans. (b* -f W)rc. 

2. Show that the area of a segment of a parabola cut off by any focal chord in 
terms of/, the chord length, and/, the parameter, is i/ 2 /2. 

3. Show that the area of the curve x 2 y 2 = (a — x)(x — b) is 7l{a* — b^) 2 - 

4. Show that the whole area between the curve y(a 2 -\- x 2 ) = ma 3 and the 
.r-axis is m-jta 2 . 

5. Show that the whole area between the curve y 2 (a 2 — x 2 ) = ft and its 

asymptotes is 2itb 2 . 

6. Show that the area between the curve and the axes in the first quadrant for 
(x/a)i -f- ( y/bf = 1 is ab/20. 

7. Show that the area of a loop of the curve y 4 — 2c 2 y 2 -f- a 1 * 1 = o is 2f r 3 n a ^ 

8. The locus of the foot of the perpendicular drawn from the origin to the tan- 
gent of a given curve is called the pedal of the given curve. 

(1). The pedal of the ellipse (x/a) 2 -f (y/b) 2 = 1 is 

p2 = a 2 cos 2 + b 2 sin 2 0. 



240 APPLICATIONS OF INTEGRATION. [Ch. XX. 

Show that its area is \-n{a 2 -f- b 2 ). 

(2). The pedal of the "hyperbola {x/af — (y/b) 2 = I is 

pr = a 2 cos 2 Q — b- sin 2 9. 

Show that its area is ab -\- (a 2 — b 2 ) tan— *(a/b). 

9. If y x , y 2 , y s be three ordinates, y 2 being midway between y x and_y 8 , of the 
curve 

y = ax 3 -\- bx 2 -\- ex -j- d, 

show that the area bounded by the curve, the .r-axis, and the ordinates y x and y % is 

If we transfer the origin to x 2 , o, and put x x = — h, x 2 = + /*, the equation 
of the curve can be written 

y = ax 3 + fix 2 4- yx + 8. 

We have for the area 

,+k 



J ydx = 2h(\ph 2 + d), 



and \h(y x +>' 3 + 4y 2 ) nas this same value. This is called Newton's rule. 

10. Show that the area of any parabola 

y — ax 2 -+- bx -f- c, 

from x — — h, to x — -f- /i, can be expressed in terms of the coordinates x l , y x 
and x 2 , y 2 of any two points on the curve, whose abscissae satisfy x x x 2 — — £// 2 . 

Ans. A = Zh * 1 ** ~~ ** y \ 

x x — x 2 

The mean ordinate in the interval is 

Let/ and q be two undetermined numbers. Then 

P>\ + qy<r-ym = a{p X 2 + qx 2 - \h 2 ) + b{px x + ?*„) + (/ + q - l)€. 

The three equations in/, <7, 

^ + qx 2 = \h\ (i) 

/■*1 H-/^2 = °> ( 2 > 

P + 9 = -ii (3> 

give determinate values of / and ^, provided 

•*"i 2 , * 2 2 > W = °» 
x 1 , * 8 , o 
i,i,i 

or jr x jr 2 = — \h 2 . 

Theri ;'« = p)\ + qy 2 » 

and the values of / and q from (2), (3) give the result. 

11. In Elliott's theorem, § 164, (3), show that the mean of the areas of the curves 
described by all points on the segment P X P 2 is ^[(^i) + (^)J — \{^)- 

12. A given arc of a plane curve turns, without changing its form, around a 
fixed point ;n its plane; what is the area swept over by the arc? 



Art. 164.] ON THE AREAS OF PLANE CURVES. 241 

13. If a curve is expressed in terms of its radius vector r and the perpendicular 
from the origin on the tangent/, prove that its area is given by 

I /* pr dr 

2 J tf-przrp' 

14. Lagrange's Interpolation Formula. 

We have seen, in the decomposition of rational fractions, that when 

tp(x) = (x - a t )(x - a 2 ) . . . (.r - q„), 
and F\x) is a polynomial in x of degree less than n, 

F(x) __ « I F(a r ) 



See § 133, and Ex. 79, Chapter XVIII. 

If F\x) is any differentiable function of x, then, since 

vanishes at x = ^ x , . . . , # M , and the second term is a polynomial of degree n — I, 
we have, § 98, II, lemma, 

where r is some number between the greatest and least of the numbers x, 
a x , . . . , a n . 
The formula 

is called Lagrange's interpolation formula. The member on the left computes the 
value at x of an unknown function when its values at a x , . . . , a n are known, 
with an error which is represented by 

(x - a x ) . . . (x>-a») ^ 
n\ v 

15. Gauss' and Jacobi's theorem on areas. 

If F\x) is any polynomial of degree 211 — 1, then the exact area of the curve 
y = Fix) between x = p, x = q can be computed in terms of n properly assigned 
ordinates. 

Let 

^ X) ~ ~ x-a r f'(a r )> 

where, as in Ex. 14, ip(x) = (x — a x ) . . . (x — a n ). 

Then J{x) = F(x) — L(x) is a polynomial of degree 2n — I, in which F[x) 
is of degree 2« — 1, L(x) of degree n — 1. Also, J{x) vanishes when x = a v 
. . . , a n . Hence 

F(x) - L{x) = A 0(x) iP(x), 

where A is some constant and (p{x) some polynomial of degree n — I, since ip(x) 
is of degree n. 

Integrating between p and q, 

r Fx) dx - r L(x)dx = a r ^{x) ^) dx. 

Jp Jp Jp 



242 APPLICATIONS OF INTEGRATION. [Ch. XX. 

Jacobi has shown as follows that we can always assign a x , . . . , a n , so that 



I 



<p ip dx ■=. o. 



For, integrating by parts successively, 

'<pi!><ix = <pi\ - 0'ip. 2 + . . . - (- i)«0(«-O^ 



/' 



where cp( r ) denotes the result of differentiating <p r times, and rp r the result of 
integrating ip r times, remembering that <p( n ~ l ) is a constant. 

It we take, after Jacobi, for the values a x , . . . , a n , the n roots of the equa- 
tion of the «th degree 



{dx) H ^ X - P){X - ?)]H = °' 



then the integrals ip x , . . . , ip n between / and q are all o, since each contains 
(x — p)(x — q) as a factor. 

Therefore, for these values of a x , . . . , a n , we have 

i"F(a r ) if>(x) 



f\x)dx= f*2p 



)* 



or the proposition is established.* 

If the degree of F(x) is 2«, then the area can be expressed in terms of n -f- I 
ordinates taken at the roots of 



/ d \ »+* 

\dxj [{X ~^ X ~ ^ M+I = °- 



The area of y = 7^x) can be expressed in a singly infinite number of ways if 
one more than the required number of ordinates be used, in a doubly infinite 
number of ways if two more than the required number be used, and so on. 

16. Show that the area of 

y = a o + a \* + a -2 x ' 2 + a z^ > 
from — h to -f h, is equal to 

where y x and y 2 are the ordinates at .v = ± A/4/3, Give a rule and compass 
construction for placing these ordinates. 

*See Boole's Finite Differences, p. 52. 



CHAPTER XXI. 



ON THE LENGTHS OF CURVES. 



Rectangular Coordinates. 



165. Definition of the Length of a Curve. — A mechanical con- 
ception of the length of a curve between two points on it can be 
obtained by regarding the curve as a flexible and inextensible string 
without thickness, which when straightened out can be applied to a 



straight line and its length measured. 



The curvilinear segment is 



then said to be rectified. 

The rigorous analytical definition of a curve and of its length is 
a more difficult matter. 

If y is a function of x such that y, Dy, D 2 y, are uniform and 
continuous functions in an interval x = a, x = /?, then the assem- 
blage of points representing 

y =/(x) 

in (a, f3) is called a curve. 

We can demonstrate * that if P and P are any two points on this 
curve, we can always take 
P and P x so near together 
that the curve between P 
and P lies wholly within 
the triangle whose sides are 
the tangents at P and P x 
and the chord PP V And 
also, if Q, R be any other 
two points on the curve 
between P and P lt then, 
however near together are 
Q and R, the same property 
is true for Q and R. 




Fig. 87. 



If we divide the interval (a, b) into n subintervals and at the 
points of division erect ordinates to the points A, L, . . . , B, etc., 
on the curve, then draw the chords through these points, and the 
tangents to the curve there, we shall have two polygonal broken lines 
ALMNB inscribed, and ATRSVB circumscribed, to the curve AB. 



Appendix, Note II. 



243 



244 



APPLICATIONS OF INTEGRATION. 



[Ch. XXL 



Let c r represent the length of the rth chord, and t r that of the rth 
side of the circumscribed line. 

Clearly, whatever be the manner in which (a, b) is subdivided or 
to what extent that subdivision be carried, we shall always have 

2c r < 2f r and £ 2c r — £ 2t = o. 

I I «=oo I n = x> i 

If we interpolate more points of division in (a, b), then 2/ 
decreases while 2c increases. Consequently 2t and 2c converge 
to a common limit. This limit we define to 
be the length of the curve between A and B. 

166. Let P be a point x, y on a curve, 
the length of which between A and P is s. 
Let P be a point on the curve having 
coordinates x -f- Ax, y -f- Ay, and let the 
length of the curve between P and P x be 
As, the length of the chord PP X be zfc. 
Draw the tangents at P and P v Let the 
angle which PT makes with Ox be 0, and 
the angle between TP and 77^ be J0. 
/ 1? then, by § 165, 

Ac < As < / -f / x . 

But, from the triangle ^77^, 

(Jc) 2 = / 2 -f /j 2 -f 2// x cos A6, 

— (/4-/J 2 - 4#, sin 2 \A6. 

Ac V 4^i 




' + 4 



(' + «' 



//#. 



4// x /(/ -j- Z^ 2 can never be greater than 1, and when A6( — )o, 
Ac\ — )o, t -f- / x (=)o, also sin 2 \A6{ — )o. Therefore when Ax( = )o > 
we have 

Ac \ 

A*( = )o 

Since, by definition, As lies between Ac and / -f t x 

'Ac 



£ 



we 



also have 



Now, 



or 



£Ac _ 

A.r( = )o 

{Acf = (Ax)* - 

j'Acy _ (Ac\ /As 
\Ax) ~~~ \Js) \Ax 



(4>') 2 > 



)'~+® 



Art. 166.J ON THE LENGTHS OF CURVES. 245 

Therefore, in the limit, for Av( = )o, 

or 

* = V I + ($)'*■ (I) 

Hence the length of the arc of the curve from A to P is 
In like manner, using 4>' instead of Ax, we obtain 

-f'A^m*- » 

In differentials 

<fr 2 = </jtr 2 + </y 2 . 

Since dy/dx = tan #, # being the slope of the curve at x, y, we 
have 

dx — cos # ds, dy — sin # ds. 

Therefore — , -7- are the direction cosines of the tangent to the 
as as 



curve at x, 


y> 


















EXAMPLES. 






1. Rectify 


the 


semi-cubical parabola ay 2 = 


:*3. 






Here 


y 


x% dy 
a i' ' ' dx 


=16)' 


ds 
dx 


-(■ 


+5 




s 


-rhz) 


'*-ll 


h 


gx\i 
~4a) 


-i 



the arc being measured from the vertex. This was the first curve whose length 
was determined. The result was obtained by William Neil in 1660. 

2. Rectify the ordinary parabola y % = 2ax. 
We have DyX = y/a. 

1 /•y , 

.-. s=- Ya*+y*dy, 
a t/0 

1 1-7. 1 a , y -\- Vy 2 4- <* 2 

the arc being measured from the vertex. 

I - -*) 

3. Rectify the catenary y — \a \e <* -f- ^ a / . 

We have Dy — \\e a — e a ) . 

.-. s = \ ( X \e~" -f *~ «/ <£r = ^a V s - * a / • 



246 APPLICATIONS OF INTEGRATION. [C11. XXL 

Show that s = PL Fig. 73. Also, NL = constant. The catenary is therefore 
the evolute of the tractrix represented by the dotted line in the figure. 

4= Rectify the evolute of the ellipse 

(ax)* -f- (by)* - (a 2 - b 2 )*. 

Write the equation in the form 



©'+» 



put x = a sin 3 0, y = (5 cos 3 0. 

. •. s = 2 3 f (a 2 sin 2 -f ft 2 cos 2 <pfid(a 2 sin 2 + /J 2 cos 2 0) . 

2(a — p-) J 

Measuring the curve from x — o, y = (5, we get 

_ {a 2 sin 3 + /5 2 cos 3 0)2 _ ^ 
~ a 2 - /i 2 

5. Find the length of the curve gay 2 = x(x — 3a) 2 , from x = o to jt = 3^7. 

6. Find the entire length of the hypocydoid xt -f y% = «*. /tfwj. 6a. 

7. Find the length of the arc of the circle x 2 -\- y 2 = a 2 , from x = o to x = b y 
and the whole perimeter. 

8. Find the length of the logarithmic curve y = ca x . 
We have D y x — b/y, where b = i/log a. 

... s = p?y* <'■ =(*>+ -•")'+ ' "* <£ + y*-\ 

9. Find the length of the tractrix (see § 163, Fig. 84) 
dy 



fdy 
— a I — = — a log 



y -f- const. 



Measured from the vertex, x = o, y = a, 

S = a log (a/y). 

x 2 v 2 
10. Length of an arc of the ellipse —-)- — = 1. 

Put .r = a sin 0, y =.- b cos 0. Then 



5 = /% 2 cos 2 -f <* 2 sin 2 0)^/0 r 
= a J (1 — e 2 sin 2 0) </0. 



where <• is the eccentricity. This is an elliptic integral and cannot be integrated 
in finite elementary functions. The length of the elliptic quadrant corresponds to 
the limits = o, (p = \n. Since e 2 sin 2 is always less than I, we can expand 
the radical in a series of powers of sin 0, and integrate, obtaining the length of the 
quadrant (see Ex. 27 § 149) 



ltd \ 
2 \ 



i(lV<1 _ (L3Y* _ /'•3-5\'< 5 _ 1 

1I2/ i V2.4/ 3 \2.4-6/ 5 



Art. 167.] 



ON THE LENGTHS OF CURVES. 



247 



Polar Coordinates. 

167. If p = /(B) is the equation to a curve, and p, 6 are the 
coordinates of P ; p -f Ap, 6 -f- A 6, those of P x , then, calling Ac 
the chord PP X , we have 

(Acf = (p + //p) 2 + P 2 - 2 P (p -f Jp)cos Z/0, 
= (zJp) 2 4- 2 p(p -f Jp)(i - cos Ad). 
Hence 

Z/c\ 2 /ApV~ . 1 _ cosz/0 

^) = U^J + 2p ( p + Jp ) ( j^» • 

But when A0( = )o, we have Z/p( = )o, Jc( = )o, and 

/i — cos x __ 1 /"sin jf _ 1 



M = )o 



= U) + p • 



or 




Also, -r>, — r = 



Jj Ac _ zfc 

A'e~As~~A~d' *' d6~dd 

... <&2 = ^ p 2 _|_ p 2 ^ 



ds dc . ds nss 

= -^ , since — = 1, § 166. 



dc 



-r>+^) , -r^+^)'* 

Otherwise we can deduce the formula for the length of an arc in 
polar coordinates directly from the corresponding formula in rect- 
angular coordinates. - — . 
For x = p cos 6, y = p sin 0. 
. •. <£r = dp cos — p sin 6 dd, dy = dp sin -{- p cos </#. 
. ■. <fr2 _ ^2 _|_ dy i = ^2 _|_ p 2 ^2 # 



APPLICATIONS OF INTEGRATION. 



[Ch. XXI. 



EXAMPLES. 

1. Find the length of the cardioid p = a(l -f cos B). 

I) p — — a sin 0, and therefore 

s = a /'[(I -f- cos Of -f- sin 2 0]" dd = la I cos $6 dB = 4 a sin IB 

The entire length is 8a. 

2. Show that the length of the arc of the spiral of Archimedes, p = aB, from 
the pole to the end of the first revolution, is 

a\n \/i -f- 47T' 2 + \ lo g ( 27t -h V 1 + 47T 2 )]- 

3. Logarithmic spiral p = a*. Put £ = i/log a. 

Then J = P (i + J 2 )* ^p = (I + ^ 2 )V 2 - >i). 

4. Show that the length of the arc of log p = aB, from the origin to (p, 0), is 

5. Find the length of p -f 2 = a 2 , 

6. Find the length of p — a sin 0, from o to \it. 

7. Find the length of p = a sec 6, from o to ±7t. 

8. Show that the entire length of p = a sin 3 \B is f #£. 

9. Show that the entire length of the epicycloid 

4(^2 _ fl 2)3 _ 27a i p 2 sin 2 0, 
which is traced by a point on a circle of radius \a rolling on a fixed circle of radius 
a, is \2a. 

10. Find the entire length of the curve p = a sin 20. 

11. Show that the length of the hyperbolic spiral pB = a is 



[ 



yd 1 -\- p l — a log 



a + \/a* + p* 



P _JP 2 

12. Show that the length of the parabola p = a sec 2 iB, from 8 = — \it 

to = 4- i7r, is 2rt(sec \it 4- log tan \n). 

168. Geometrical Interpretations of the Differential Equations 
ds' 2 = dx 2 + ^/ ! and <ft a = dp 1 4- pW 2 . 

I. In Cartesian coordinates, if we take x 
as the independent variable, then we have 
PM = dx. Also, since 

Dy = tan B = tan MPT, 
. •. </r = tan B dx - MT. 
. -. </.r 2 = /W 2 4- 3/r 2 = PTK 
Hence ds = PT, while the correspond- 
ing difference of the arc is PP V Also, we 
have the relations 

dx — ds cos 0, dy — ds sin 0. 
It is easily shown geometrically that the 
limit of the quotient of ds = PT, by either 
Fig. 90. /} s — pp x or Jr = PP V is 1, when dx = Ax 

converges to O. Set- definition of the length of a curve, § 165. 









T 


y 


T 


>/y^ 


M 








/B 


dx 














Art. iC8.] 



ON THE LENGTHS OF CURVES. 



-49 



II. We can in the case of polar coordinate 
lengths of certain circular arcs as follows: 

Let OA be the initial line and P a 
point on the curve /(p, 0) = o, PT the 
tangent at P. Draw OC perpendicular 
to p = OP, cutting the normal at P / 

in C. Then n = PC is the normal 
length, and S, t = OC is the subnormal. 
Let 9 be the independent variable, then 
d r j — AQ is an arbitrarily chosen angle. 
We have the differentials ds, dp, p dQ 
proportional to the sides of the triangle 
POC, or to n, S„, p, respectively. For 
we have 



ixhibit ds, dp, and p dO as the 



@°= & 



+ P* 





But dp = S n dQ, by §92, (5). Also, S n 2 + p 2 = n 2 . Hence ds = tide. 
Draw .PC 'parallel and equal to OC. Strike the arcs PN, PL, and PM with centers 
C radius S n , C radius n, O radius p, having the common central angle AQ — dQ. 
Then 

ds = PL, dp = PN, pdQ = PM. 

It is interesting to notice that the rectilinear triangle PLNis a right-angled tri- 
angle similar to PCO; the sides of which, PL, PN, NL, are equal to the chords 
jr subtending the arcs PL, PN, and PM respectively. 

Therefore, in the triangular figure PLN whose sides are 
the circular arcs PL, PN, and an arc LN with, radius p equal 
^ to PM, ;we have the sides {PN) = dp, (PL) = ds, (LN) — p dQ. 
Also, the angles between the circular arcs are 

(Z) =&- if>, (P) = if>, (N) = i7t+ dd. 
and ds 2 = dp 2 -+- p 1 dQ 2 , 

dp = ds cos ip, p dQ = ds sin zl: 
In order to prove these statements, it is only necessary to 
show that the rectilinear segment LN is equal to the chord 
subtending PM. Let x, y be the chords subtending PL, PN. Then from the 
rectilinear triangle PNL we have 

LN 2 = x 2 -\-y 2 - 2xy cos LPX. 
But / LPN = iff -f \AQ - \AB = if). 

Also, x = 2n sin \AQ, y = iS n sin \AQ. 

.-. LN 2 — 4(S n 2 + ri 2 - 2nS n cos rj>) sin 2 \AQ, 
= 4p 2 sin 2 \AQ = (chord MP) 2 . 

The remainder follows easily. 

Observe that if we draw PM perpendicular to OP, as 
in Fig. 93, and put PM = 8p, MP X = dp, then we have, 
for AQ(=)o, 

jTa^ _ r sp _ t rsp _ 
X*- z \ JL^~ l'^- 1 ' 

Therefore the difference equation 
Ac 2 = dp 2 -\- dp 2 
leads at once to the differential equation 

ds 2 = dp 2 + p 2 dQ 2 . 




250 



APPLICATIONS OF INTEGRATION. 



[Ch. XXI. 



169. Radius of Curvature and Length of Evolute. 
If/" (Xj r) = o is the equation of a curve, then 



sec 2 6> 



dy f a d\y 
Hence if R is the radius of curvature at 



dd 



dx' 



R 



(1 +y 2 r 



a dx 

sec v — - n = 



ds 

dd 1 




y dd 

since ds = sec 6 dx. Therefore ds = R dd. 

The angle A6 = dS is the angle between the tangents at P and P v 
and is equal to the angle between the normals at P and P x . 

170. The length of the arc of the evo- 
lute of a given curve is equal to the differ- 
ence of the corresponding radii of curvature 
of the involute. 

Let x, y be a point on the involute cor- 
responding to the point a, ft on the evo- 
lute. 

Then we have for the radius of curvature 

R 2 = («-*)* + (ft -J') 2 - 

Fig. 94. Differentiating, we get 

R dR = (a - x)(da - dx) -f- (ft -J')(dft - dy), 

= (a-x)da + (ft-j>)dft, (1) 

since (a — x)dx -\- (ft — y)dy — o, this being the equation of R, 
the normal to the involute. If 6 is the angle which the tangent to the 
involute at x, y makes with Ox, then, since R is tangent to the evolute, 
R makes with Ox the angle <p — \n + 6, and we have 

a — x = — R sin 6 = R cos 0, ft —y = R cos = R sin <p. 
Hence, on substitution in (1), 

dR — da cos -f- dft sin 0, 
= da, 
if cr is the length of the arc of the evolute. 

Integrating between a lt ft x and a v ft 2 , we have 

R 2 -R, = cr 2 - <T X . 
This can be shown otherwise, for we have 

(x — a)da + (y — /?)<//? = o, (2) 

the equation to the normal to the evolute at a, ft. 

The perpendicular R, from x,y on the involute to the line (2), is 

^ = ( x-^)^-f (y-ft)dft 
t/doP+dftP 
or (a- - ff)fl?« + (y - ft)dft = R da. 

Equating with (1), we ge f dR = f/cr as before. 



Art. 171. J ON THE LENGTHS OF CURVES. 251 

It is to be particularly observed that the theorem as enunciated ap- 
plies only to an arc of the involute such that between its ends the 
radius of curvature has neither a maximum nor a minimum value. 
For when JR passes through a maximum or a minimum value dR 

changes sign. / dR would be zero when taken between two points 

at which R has equal values. 

In applying the theorem one should be careful to determine the 
maximum and minimum values of the radius of curvature for the invo- 
lute, and add the corresponding absolute values of the lengths of the 
evolute, when the radius of curvature has maximum or minimum values 
between the ends of the arc under consideration. 

From a mechanical point of view, since the evolute is the envelope 
of the normals of the involute, we can regard the involute as a point 
described by a point in the tangent, as the tangent is unrolled from 
its contact with the evolute ; the arc being considered as a flexible 
inextensible string wrapped on the curve. The truth of the above 
theorem from this point of view is made evident. 

The theorem of this article rectifies any curve which is the evolute 
of a known curve whose radius of curvature can be found. 



EXAMPLES. 

1. Find the length of 27 ay 1 = $(x — 2a f, the evolute of the parabola y 2 = \ax. 
We have for the coordinates x 1 , y' of the center of curvature and R, the radius 
of curvature of the parabola at x, y, 

v 3 (a + x\ * 

Measuring the arc of the evolute from the cusp, x = 2a, y = o, to x J , y\ 
we have 

Ir' -I- a\ 3 

S 



/V + a\ * 

= 2a ( I — 2a. 

\ 3* J 

2. Find the length of 

(ajc)i + (byf = (a 2 - <5 2 )§, 
the evolute of the ellipse. 

3. Show that the catenary is the evolute of the tractrix, and find the length of 
an arc of the catenary as such. 

171. The Intrinsic Equation of a Curve. — The length s measured 
from the point of contact of a curve with a fixed tangent, and the 
angle <p which the tangent at the end P of the arc makes with the 
fixed tangent, are called the intrinsic coordinates of a point on the 
curve. 

The equation /[ s, <fi) = o, which expresses the relation between s 
and 0, is called the intrinsic equation of the curve. 

To find the intrinsic equation of a curve /(x,j') = o or f(p, 0)=o, 
we have to find the length of the arc from a fixed point to an arbitrary 



252 APPLICATIONS OF INTEGRATION. [Ch. XXI. 

point on the curve, then the angle between the tangents there. 
Eliminating the original coordinates between these three equations, 
the result is the intrinsic equation. 

EXAMPLES. 

1. Find the intrinsic equation of the catenary. 
Take the vertex as the initial point, then 

y = \a\^ + T«) . 

I- —\ 

Dy = % \e a — e a J = tan 0. 

Also, s = \a \e a — e a J . 

Eliminating x, we have s = a tan (p. 

2. Find the intrinsic equation of the involute of the circle. 

Let x 2 -f- J 2 = « 2 he the equation to the circle. Unwrap the arc beginning at 
the point a. o, and let the radius to the point of contact make the angle <p with 
Ox. Then is also the angle which the tangent to the involute makes with Ox. 
The radius of curvature is the unwrapped tangent length, or J\ = a<p. But 

ds = R d<p = a<p d<p. . •. s ■=. ^a<p' 1 . 

172. General Remark on Rectification. — The problem of find- 
ing the length of any curve whose equation is given involves the 
integral of a function which is in irrational form. This in general 
does not admit of integration in finite form, and cannot generally be 
expressed in terms of the elementary functions. There are, generally 
speaking, but few curves that can be rectified, in terms of elementary 
functions. 

EXERCISES. 

1. Show that the length of Sa 3 y — x* -f- ba' 2 x 2 measured from the origin is 
x(x 2 + 4a 2 )*/8a s . 

2. Show that the whole length of 4(x 2 -\- y 2 ) — a 2 = 3« 3 ;'^ is 6a. 

3. Show that a'"y n = x m + n is a curve whose length can be obtained in finite 

. n n I . 

terms when — or 1 is an integer. 

2W 2m 2 

4. Show that the intrinsic equation toj 3 = ax 2 is 

s = -^a (sec :1 — 1). 

5. Show that p m = a m cos ?u0 can be rectified when \/m is an integer. 

6. If x — <p{t\ y = ^(/), then 

7. In the cycloid x — a(0 — sin 0), y = a vers 0, show that 

ds = 2a sin \Q dQ. 
Hence the length of one arch is 8rt. 



Art. 172.] ON THE LENGTHS OF CURVES. 253 

8. Show that the length of the cycloid 

x — a cos— 1 — -\- \l2ay — y' 1 

a 

from the origin to x, y is \/8ay. 

9. Show that the intrinsic equation of the cycloid in Ex. 8 is 

s = \a sin 0, 
the tangent at the origin being the initial tangent. 

9 

10. In the equiangular spiral p = ae * . show that s = p sec ip, where tan if; 
= n, measuring s fiom the pole. 

11. Find the length of the reciprocal spiral from = 2it to = ^7t, the equa- 
tion being pb — a. 

12. Show that the whole perimeter of the lemniscat: 



4M" + A+^+ I - 3 ' 5 



2.5 2-4-9 2-4-6-13 ' J 

13. Show that the length of y 2 = x 3 between x = o, x = 1 is ^ T (132 — 8). 

14. Designating by Z*I* (/) the length of the curve f{x, y) = o, from x — a 
to x ■=. b, show that: 

(a). L x x l\{x* - 6xy + 3) = ft. 
(3). L x x Z a (xl + y$ - J) = 6a. 
(c). L x x \\\_2y — x |/x 2 — 1 — log(x — |/x 2 - 1)] = / x dx= 4. 



(^). /?;! 



x — ya 2 — y l -j- a log — j = a log — . 



{e).L x x = = \l{y - log sin ^) = log 3. 

(/). L x x Vo(y - V x - x2 - sm_I V*~) = Px-^dx = 2. 
(g). L y y Z\{y 2 + 4r - 2 log/) = £(2 log 2 + 3). 

15. Use the binomial theorem to evaluate 

t/ a x 

16. Show that 

LZ2l(y - sm *) = «(l + i - A + l§¥ - • • • ) 

17 - Z £:*V - ^ + « 2 ) = ^ + log 4). 

20. A hawk can fly z/ feet per second, a hare can run v' feet per second. The 
hawk, when a feet vertically above the hare, gives chase and catches the hare when 
the hare has run b feet. Find the length of the curve of pursuit. 



: 54 



APPLICATIONS OF INTEGRATION. 



[CH. XXI. 



Take 0, the starting-point of the hawk, as origin, the line OH drawn to the 
starting-point of the hare //as j-axis, and a parallel Ox to the hare's path as x-axis. 
When the hawk has flown a distance s to P, the hare will have run a distance a to 
P' in the tangent to the curve at P. 



y 




P' 




j i 




v)*^^- — 


' C 


a 




t£ 




u 









Fig. 95. 

Let PP' = t. We have b — v' T- HC. S = vT, the length OPC. T being 
the time of pursuit. 

b v' 



In like manner, 



— = — = k = constant. 
S v 



z=* = k. 



S V 

da = k ds. Also, 

f 2 = (o- - xf -j- {a - yf. 
Idt = (a — x)(d(T — dx) — [a — y)dy. 
But a — x = t cos 0, a — y = t sin 9. Also, dcr cos 6 is k ds cos 6 or 



k dx, and 



Hence 



dx cos -(- </}' sin 6 = <&. 
. •. dt — k dx — ds. 



pb /»0 

S = k \ dx - dl = kb + a, 

5- + -J4+*- 






21. If Z 1 and rare the radii of the fixed and rolling circles in the epicycloid and 
hypocycloid 

x = (P ± r) cos d> T r cos <p, y = (P ± r) sin T r sin (p, 

± r ± r 

show that the lengths of the curves from cusp to cusp are respectively 

Sr(P ± r)/R. 

22. In § 164 (Elliott's theorem), if s x , s 2 s are the corresponding lengths of 
the arcs described by the points P x , F 2 , P respectively and a the corresponding 
length of the director curve, show that 

ds* = k % ds* + l\ ds* - k x k % da'K 



CHAPTER XXII. 
ON THE VOLUMES AND SURFACES OF REVOEUTES. 

173. Definition. — A point is said to revolve about a straight line 
as an axis when it describes the arc of a circle whose plane is per- 
pendicular to the straight line and whose center is on the straight 
line. 

A plane figure is said to revolve about a straight line in its plane 
as an axis when each point of the figure revolves about the line as an 
axis. 

The solid geometrical figure generated by the revolution of a 
plane figure about a straight line in its plane as axis is called a 
revolute. The surface of the figure revolved generates the volume, 
and the perimeter of the figure revolved generates the surface of the 
revolute. 

Examples of revolutes are familiar from the three round bodies 
of elementary geometry, the cylinder, cone, and sphere. 




Fig. 96. 



AB being the axis of revolution, the cylinder is generated by the 
revolution of the rectangle ABCD, the cone by that of the right- 
angled triangle ABC, the sphere by that of the semicircle APB. 
The volumes of the revolutes are generated by the surfaces, and the 
surfaces of the revolutes by the perimeters of the revolving figures. 

We know from elementary geometry that the volume of the 
cylinder is equal to the area of the circular base multiplied by the 
altitude or by DC the length of the line generating the curved surface. 
Also, the curved surface of the cylinder has for its area the product 
of the circumference of the base into CD, the length of the generating 
line. 



256 



APPLICATIONS OF INTEGRATION. 



[CH. XXII. 



It is evident from the definition of a revolute that any section of 
a revolute by a plane perpendicular to the axis AB is a circle, such 
as ODD' . The circular sections cut out of the surface by planes 
perpendicular to the axis are called parallels. In 
like manner the section of the surface of a revolute 
by any plane passing through the axis is a line 
identically the same as the generating line. For if 
in the figure the surface is generated by the revolu- 
tion of the line ACDB about the axis AB, then the 
section AD ' B is nothing more than one position of 
the generating line ACB. Again, the revolute can 
always be regarded as being generated by a circle 
moving in such a manner that its center moves 
along the axis to which its plane is perpendicular, 
and its radius changes according to a given law. 




174. Volume of a Revolute. — Let y =f(x) be a curve AB. 
We require the volume of the solid generated by the figure aABb 
revolving about Ox as axis ^b 

of revolution. 

Divide (a, b) into ;/ 
subintervals, and pass 
planes through the points 
of division cutting the 
solid into n parts, such as 
the one generated by the 
revolution of xPP'x'. 
We can always take x' — 
x = Ax so small that the 
curve PP' will lie inside 
the rectangle PMP'M', if 
f(x) is continuous. Let 
y be an increasing one- FlG - 9 8 - 

valued function from x = a to x = b. The volume, A V, of the 
solid generated by xPP'x ', lies between the volumes of the cylinders 
generated by the rectangles xPM'x' and xMP'x ' . Hence for each 
subdivision of the solid we have 




ny 2 Ax < AV < tty'^Ax. 



W 



The whole volume of the revolute, therefore, lies between the 
sum of the n interior cylinders and that of the exterior cylinders, or 



2 ny' l Ax < V < 2 7ry' 2 Ax. 



(*) 



Hut if we interpolate more points of subdivision in (a, b), we increase 



Art. 174.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 257 

the sum of the interior volumes and decrease that of the exterior; and 
since 

TC&Ax 



I 



these sums have a common limit, which is V. 

Ax( = )o « 

' ■'• V x = £ STryVx, 



I Ttfdx. (3) 



Again, we have directly from the inequality (1) 

ny l < — — < ny *. 

Hence, for zLv( = )o, we have 

dV 2 

since £y' = v, when Ax( — )o. 

.-. dV=ny' i dx, (4) 



and as before 



ab 

n I jP dx. 



(5) 



In like manner we show that if x is a one-valued function of y, 
say x = 00')' th en tne volume generated by the revolution of the 
curve about Oy as an axis, included between two planes perpendicular 

to Oy at y = p and y = g, is 

V y =itf p x*dy. (6) 

EXAMPLES. 

1. Find the volume of the cone of revolution generated by the revolution of the 
triangle formed by the lines x = o, y = o, x/a -f- y/b = I, about Ox as axis. 

v r - . /•« /. 






But « is the altitude and ^ the radius of the base of the cone. Therefore the 
volume is equal to one third the product of the area of the base into the altitude. 

2. Find the volume of the sphere generated by the revolution of a semi-circle 
of x 2 -f- y' 1 = a' 1 about Oy. 

V y = n f a x 2 dy = it f a (a 2 —y 2 ) dy - *7ta*. 

J —a J — a. 

3. The prolate spheroid is the revolute generated by the revolution of an ellipse 
about its long axis, sometimes called the oblongum. 



25^ 



APPLICATIONS OF INTEGRATION 



[Ch. XXII. 



Let a be the semi-major axis of x 2 /a 2 -j- y 2 /b 2 ■. 
Then we have for the volume of the oblongum 



/' 



-a fl 



4. The oblate spheroid or oblatum is the revolute obtained by revolving the 
ellipse about its minor axis; show that its volume is ^7ta 2 b, where b is the semi minor 
axis. 

5. Show that the volume of the revolute obtained by revolving the parabola 
y 2 = \ax about Ox, from x = o to x = a, is 27ra'\ 

This is the paraboloid of revolution. 

x 2 y 2 

6. If the hyperbola — — — = I revolves about Oy, the revolute is called the 

a 1 b l 

hyperboloid of revolution of one sheet. Show that the volume fromj' =o \o y =y 

is^jV + 3^)- 

If the curve revolves about Ox, find the volume from x = a to x = 2a. This 
surface is called the hyperboloid of revolution of two sheets. 

7. Find the entire volume generated by the revolution about Ox of the 
hypocycloid x^ '' -\- y^ — aK Ans. fifelta 9 . 

8. The surface generated by the revolution of the tractrix about its asymptote is 
called the pseudo-sphere. This important surface has the property of having its 
curvature constant and negative. Find its volume. 

Here y 2 dx = — {a 2 —y 2 )^y dy. Hence the volume from x = o to x = x is 



V, 



=*/ v 



y 2 f.y dy = \n{a> - y 2 f 



The volume of the entire pseudo-sphere is %rta z , or one half that of a sphere 
with radius a. 

9. Find the volume generated by the revolution of the catenary 

I - -A 

y = %a\e a -\- e a J about Ox from O to x. Ans. \ita{ys -J- ax). 



10. The volume generated by revolving the witch (x 2 -f- \a 2 )y 
asymptote is 4^r 2 « 3 . 



8a 3 about its 




Fig. 99. 



175. To find the volume of the 
revolute generated by a closed curve 
revolving about an axis in its plane, but 
external to the curve. 

We take the difference between the 
volumes of the revolutes generated by 
MABCN and MADCN. Hence the 
volume of the solid ring generated by 
ABCD revolving about Ov is 

where x x = RD, x 2 = RB, and the 
limits of the integral are y = OM, 



Y = ON. A corresponding integral gives the volume about Ox* 



Art. 176.J OX THE VOLUMES AND SURFACES OE REVOLUTES. 259 



EXAMPLES. 

1. The solid ring generated by the revolution of a circle about an axis external 
to it is called a torus. Show that the volume of the torus generated by the circle 

(x - fl)»+y = r 2 
(a £^ r) about Oy is 2X 2 r 2 a. 

We have x. 2 = a -\- |/r 2 — y 2 , 



x x = a — |/r 2 — y*. 

V y = 7t \a 4/r 2 — y 2 dy — 2it t r l a. 

Observe that the volume is equal to the product of the area of the generating 
circle into the circumference described by its centre. 

2. Show that the volume of the elliptic torus generated by 
(f ~ ff , y _ _ 

{c > a) about Oy is 27t 2 abc. 

176. The Area of the Surface of a Re volute. — We know, from 
elementary geometry, that the curved surface of a cone of revolution 
is equal to half the product of the slant height into 
the circumference of the base. 

The area of the curved surface of the frustum 
included between the parallel planes AD and BC is i 

therefore / 

7i{VD.AD- VC-BC). / 

Since BC/AD = VC/VD, we deduce for the ^= 
surface generated by the revolution of CD about Fig. 100. 

BA the area 

2 7TMN.CD, 

where MN joins the middle points of AB and CD. 

In the figure of § 174, Fig. 98, subdivide, as before, the interval 
(a, b) into n parts; erect ordinates to the curve AB at the points of 
division. Join the points of division on the curve by drawing the 
chords of the corresponding arcs, thus inscribing in the curve AB 
a polygonal line AB with n sides. Let PP' be one of the sides 
of this polygonal line. The curved surface of the frustum of a cone 
generated by the chord Ac = PP r revolving about Ox has for its 
area 

27r yJry Ac = 27t{y + \Ay)Ac. 

We define the surface generated by the revolution of the arc of 
the curve AB about Ox to be the limit to which converges the sur- 
face generated by the revolution about Ox of the inscribed polygonal 
line, when the number of the sides of the polygonal line increases 
indefinitely and at the same time each side diminishes indefinitely. 



260 APPLICATIONS OF INTEGRATION. [Cn. XXII. 

To evaluate this limit, we have for the area of the surface gen- 
erated by the curve AB 

n = » i 
&*(=)o n 

— £ ^27iyds. 

Since for each pair of corresponding elements of these two sums 
we have 

27i(y + \Ay) Ac 



£ 



2 7ty ds 
Hence we have, by definition of an integral 



= i. 



>x = b 



/*x = o 

S x = 27t yds, 

•' x = a 

In like manner, if AB revolves about Oy, we have for the area of 
the surface generated 

/*x = b 
S y = 2 7T I Xds, 

EXAMPLES. 

1. Find the surface of the sphere generated by the revolution of the circle 

y 2 = a 2 — x 2 about Ox. 

dy x ds a 

We have — = , — = — . 

dx y dx y 

.-. S x = 27t I y ds — 27t{r 2 — x x )a. 

Hence the area of the zone included between the two parallel planes is equal 
to the circumference of a great circle into the altitude of the zone. If x 2 = -f- a, 
x x — — a, we have the whole surface of the sphere \na 2 . 

2. Show that the curved surface of the cone generated by the revolution of 
y = x tan a about Ox, from x = o to x — //, is nli 2 tan a sec a. Verify the for- 
mula deduced for the surface of a frustum in § 176. 

3. Surface area of the paraboloid of revolution. 
Let j 2 = 2tnx revolve about Ox. Then 

m j 3/// i 1 

4. Let 2a be the major axis of a 2 y 2 -f b 2 x 2 = a 2 b 2 , and e its eccentricity. Then 
wd have for the surface of the prolate spheroid 

_ 27tbe >+« \a 2 . . / , — , sin- 1 A 

s ' = - L \? - x dx = 2nah V 1 ~ ' + — ) ■ 



Art. 177.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 261 

5. Show that the surface of the pseudo-sphere is 

S x = 27ta I dy — 2iza<n — y). 



Its entire surface is 2nd 1 -. 
6. In the catenary show that 



from .r = o to x = x. 



S x = it{ys -f or), 

J^, = 27T(a 2 4- xs — ay), 



7. Show that the surface of the hypocycloid of revolution generated by 
xi -\-yi = «l about (9* is 3^-rta 2 . 

8. A cycloid revolves around the tangent at the vertex. Show that the whole 
surface generated is ^-itd 1 . 

9. The cardioid p = a(i -J- cos 6) revolves about the initial line. Show that the 
area of its surface is ^-iza 2 . 

177. If a plane closed curve having an axis of symmetry revolves 
about an axis of revolution parallel to 
the axis of symmetry and at a distance ^ 
a from it, then we shall have for the -, 
volume and surface of the revolute gen- 
erated, respectively, 

V =■ 27taA, S — 27taL, 
where A is the area and L the length of 
the generating curve. 




Fig. ioi. 



Let x =: a be the axis of symmetry 
and Oy the axis of revolution. Then 
for the volume 

Vy = 7tf q (x?-x?)dy. 

But if CM — CN = x', x 2 — a -f- x', x 1 = a — x', 
V y = 2na I 2x' dy = 271a A. 



For the surface 



S y = 27t\ (x 2 -f x x ) ds, 

= 271CL I 2ds — 27taL. 



The results obtained assume that the axis of revolution does not cut 
the generating curve. 

EXAMPLES. 

1. The volume and surface of the torus generated by the revolution of a circle 
of radius a about an axis distant c from the center (c 2^ a) are respectively 2ri l a 2 c 
and \ri l ac. 

2. The volume generated by the revolution of an ellipse, having 2a, 2b as major 
and minor axes, about a tangent at the end of the major axis is 2n l d l b. 



262 APPLICATIONS OF INTEGRATION. [Ch. XXII. 

EXERCISES. 

1 Show that the segment of the parabola f = 2 P x, made by the line 
when rotated about Ox, generates the volume 

271 p f x dx — 7tpa 2 . 

2. The figure in Ex. I rotated about the^-axis generates the volume 

3. The volume generated by the closed curve ** - aW + W = ° about the 
^r-axis is 

2 JL ['(aW -x*)dx = T %7ta\ 
a' 2 Jo 
4 The curve * 2 + >* = I rotating about the j-axis generates a solid whose 
volume is |7r. 

5. The volumes generated by y = e* about Ox and Oy are respectively 

* f° e**dx = *jr, 7T J\^zy? d >' = 27r - 

6. The curve JV = sin * rotating about Ox and 0/, respectively, for x = o, 
at = "#, generates the volumes 

at Am** 4r = \it\ it Jf{j - 2 ^) cos * <** = 27r2 - 

7 The volume generated by one arch of the cycloid 

x = a(B- sin 6), y = *(i - cos 9), rotating about Or, is 

327T<z 3 f "sin 6 |0 </(£0) = 5* 2a *- 
8. The same branch rotating about Oy gives the volume 

4? rV fV - 6 + sin 6) sin <ft - 6*r»«». 

9 Show that the whole surface of an oblate spheroid is 
/ i-e\ I -I- A 

, being the eccentricity and a the semi-major axis of the generating ellipse. 

10. The curve y\x - 4*) = «4* - 3«V &° m * = o to * = 3«, revolving 
around Or generates the volume £jr^(i5 - 10 log 2). 

11. The curve/ 2 ! 2a - x) = x* revolves around its asymptote. Show that the 
volume generated is 2it 1 d s . 

12. The curve xf = \a\2a - x) revolves around its asymptote. Show that the 
volume generated is \rt 2 a % . 

13 Find the volume and the surface generated by revolving^ = \ax about Oy, 
iroJx = otox = a. Ans. V=\**. S = ^'[e^-logfa +24/2)] . 

H Show that the volume generated by revolving the part of the parabola 
x k + ^ = J between the points of contact with the axes about Ox or Oy is ^jw . 



Art. 177.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 263 

15. The surface generated by r = x'\ from x = o to x = 1, rotating about Ox, 

is 

27T / f/l -f- gx* x*dx = — ( |/iooo — 1). 
Jo 2 7 

16. The surface generated by x* — a 2 x 2 -f 8d£y a = o, about Ox, from x = o 
to x = a, is 

7T C a 

I ($a 2 x — 2x 3 )dx = \-Jia 2 . 

\ a ~ Jo 

17. If a circular arc of radius a and central angle 2a < it revolves about its 
chord, the volume and surface of the spindle generated are respectively 

27^(1 sin a -f- i sin a cos 2 a — a cos a), 47r« 2 (sin a — a cos a). 

18. The surface generated by x* -j- 3 = 6xy turning about Oy, from x = I to 
.r = 2. has for area 7T(-\ 5 - + log 2), and ||7T when turned about Ox. 

19. The surface generated by y 2 -\- \x = 2 log y, rotating about Ox, from y = 1 
to j = 2, is -L -7T. 

20. The area of the surface of revolution of 

zy = x |/r 2 — 1 -f- log (x — 4/x 2 — 1), 
about Oy, from jr = 2 to x = 5, is 7877-. 

21. The surface of the cycloid of revolution is ^-ita 1 , and its volume is $it 2 a z , 
the base being the axis of revolution. 

22. When the tangent at the vertex is the axis of revolution, in Ex. 21, the 
surface and volume are ^-na 2 and 7r 2 tf 3 . 

23. When, in Ex. 21, the normal at the vertex is the axis of revolution the sur- 
face and volume are respectively 

$jta\it - £), itd\\ii 2 - |). 

24. Show that when the lemniscate p 2 = a 2 cos 29 is revolved about the polar 
axis, the surface generated is 

\Tta 2 I"*" sin Q dO = 27ta 2 (2 - j/T). 

25. Show that if the curve y 2 == ax 2 -\- bx -\- c be revolved about Ox, the 
volume generated between x x , x 2 is 

v* = f (*. - *i)Ui 8 + y* + 4*1). 

where jj/ w is the ordinate at -^(x 1 -f- jr 2 ). 

This curve can be made any conic whose axis coincides with Ox, by properly 
assigning the numbers a, b, c. The result then gives the volume of any conicoid 
of revolution around one axis of the generating curve. 

26. Show that the volume of the egg generated by 

xY = (x- a)(b - x), 
revolving about Ox as an axis, is 

it j (a + b) log - - 2(6 - a) I . 

27. The volume of the heart-shaped solid generated by revolving p = a(i-\-cos Q) 
about the initial line is f 7T0 3 . 

28. Find the volume of the hour-glass generated by revolving the curve 

y* — 2c 2 y 2 -f- a 2 x 2 — o about Oy. 



CHAPTER XXIII. 



ON THE VOLUMES OF SOLIDS. 



178. We have seen that the volume of a revolute is generated by 
a circular section moving with its center on a straight line and its 
plane always perpendicular to that straight line. If H \s the distance 
between any two circular sections A 1 and A 2 , and A the area of the 
circular section at a distance h from A lt then the volume included 
between the sections A x and A 2 is 



J Adh. 





§174, (3)- 

We propose to generalize this and to show that this same formula 
gives the volume of any solid included between two parallel planes 
whenever the area A of a section of the solid by a plane parallel to 
the two given planes can be determined as a continuous function of 
its distance from one of them. 

In the first place, we observe that if the plane of any plane curve 
of invariable shape moves in such a manner that 
the plane of the curve remains parallel to a fixed 
plane and the curve generates the surface of a 
cylinder, then the volume of the solid generated 
H is equal to the area of the generating curve multi- 
plied by the altitude of the cylinder generated. 
For we can always inscribe in the curve a polygon 
of n sides which will generate a prism as the 
curve moves in the manner described. If P is the 
area of the polygon and H its altitude, then PH 
is the volume of the prism. When n = <x> and 
each side of the polygon converges to o, the area of the polygon 
converges to A, the area of the curve, and the prism and cylinder 
have the same altitude H. The volume of the cylinder is the limit 
of the volume of the prism and is therefore AH. 

179. Volume of a Solid. — Consider any solid bounded by a 
surface. Select a point and draw a straight line Ox in a fixed 
direction. Cut the solid by two planes perpendicular to Ox at points 
X y , X 2 distant X x and A' 2 from 0. 

Whenever the area A of the section PM oi the solid by any plane 
PM perpendicular to Ox, distant x from 0, is a continuous function 

264 



Fig. 102. 



Art. 179.] 



ON THE VOLUMES OF SOLIDS. 



265 



of a, then the volume of the solid included between the parallel 
planes at X. and X 2 is 

V= f* % Adx. 

To prove this, divide the interval between A r x and X 2 into a large 
number of parts, n. Draw planes through the points of division 
perpendicular to Ox, thus dividing the solid into n thin slices, of 

W 




Fig. 103. 



which MPP X M X is a type. Let A be the area of the section PM, and 
A x that of section P X M X at a distance x x from 0. Let A V be the 
volume of the element of the solid included between the sections at 
x and x x , and x x — x = Ax the perpendicular distance between the 
sections. 

We can always take Ax so small that we can move a straight 
line, always parallel to Ox, around the inside of the ring cut out of 
the surface by the planes at x and x\ in such a manner as to always 
touch this part of the surface and not cut it, and thus cut out of the 
element of the solid a cylinder whose volume is less than A V. Let 
the area of the curve traced by this line on the plane PM be A'. 
Then the volume of this cylinder is 

6 V' = A' Ax. 

In like manner, we can move a straight line parallel to Ox around 
the ring externally, always touching and not cutting it. Thus 
cutting out between the planes of the sections at x and x x a cylinder 
of which the element of volume of the solid is a part. Let this 
straight line trace in the plane PM a curve whose area is A" . The 
volume of this external cylinder is A" Ax. 

Hence we have 

A' Ax < AV < A" Ax, 
or 

A' < ^<A". 
Ax 

Also, necessarily, from the manner of construction of the lines 

bounding the areas A' and A", 

A' <A < A". 



266 APPLICATIONS OF INTEGRATION. [Ch. XXIII. 

If now the surface of the solid is such that the boundary of the 
section P X M X at x 1 converges to the boundary of the section PM at 
x, when x x (=)x, then also A f (=)A, A"( = )A, and we have 

£ = '■ 

dx 
Therefore 

V= f X dV= r~dx= [ X Adx. 
Jx x J Xx dx Jx x 

d as a function o 
latter of integratio 

V = f X <p(x) dx. 



When A is determined as a function of x, say A = <p(x), then 
the evaluation of V is a matter of integration, and we have 



EXERCISES. 

1. If the parallel plane sections of any solid have equal areas, then 

V=J X ^Adx = (X 2 -X l )A. 

Therefore, if a plane figure moves in any manner without changing its area 
or the direction of its plane, the volume generated is equal to that of a cylinder or 
prism whose base is equal in area to that of the generating figure and whose alti- 
tude is equal to the distance between the initial and terminal positions of the 
generating plane. 

2. The general definition of a cone is as follows: 

A straight line which passes through a fixed point and moves according to any 
law generates a surface called a cone. In general, the cone is defined by a straight- 
line generator passing through a fixed point, the vertex, and always intersecting a 
given curve, called the directrix. 

A cone is generated by a straight line passing through 
a fixed point V, and always intersecting a closed plane 
curve of area B. Find its volume. 

Draw a perpendicular VM = H to the plane of the 
curve. Draw a plane parallel to B cutting the surface in a 
curve of area A, at a distance VN = h from V. Then we 
shall have 

A_& 

B~ H* 
Fig. 104. 

For, inscribe any polygon in the curve B and join the 
corners to V. The edges of the pyramid thus formed intersect the parallel plane 
containing A in the corners of a similar polygon inscribed in section A. If P and/» 
are the areas of these polygons, we have 

p _ W 

from elementary geometry. But A and B are the respective limits of/ and P. The 
volume of the cone is then 




= /. Adh= L B m " = **"■ 



3. A conoid is the surface generated by a straight line moving in such a 
manner as to always intersect a fixed straight line and remain parallel to a fixed 



Art. 179.] 



ON THE VOLUMES OF SOLIDS. 



267 



plane. If the generating line is always perpendicular to the fixed straightdine di- 
rector and traces a curve in a plane parallel to the directing straight line, the conoid 
is said to be a right conoid, and the curve is called its base. 

Find the volume of a right conoid having a closed plane curve of area B for its 
base. 

Let A VC be the straight-line director at a dis- 
tance VD = i/from the plane of the base. 

Any plane VNM perpendicular to VC cuts out 
of the surface a triangle of constant altitude //, and 
base MN = y. This triangle moving parallel to 
itself generates the volume required. Hence 

*AC /*AC 

V= I Tdx = / \Hy d x , 

Jo JO 

where T = area MVN, x = OD = AV. 



But (y dx = 



B, the area of the ba 
V = \HB. 



Therefore 




Fig. 105. 
is therefore half that of a cylinder on the base B 



The volume of the conoid 
having the same altitude H. 

This is at once geometrically evident by constructing the rectangle on MN as 
base with altitude H. 

4. On the oixlinate of any plane curve, of area B, as base a vertical triangle is 
drawn with constant altitude H. Show that whatever be the curve traced by the 
vertex V \\\ the plane parallel to the base, as the ordinate generates the area of the 
base, the triangle generates a volume \HB. 

5. A rectangle moves parallel to a fixed plane. One side varies directly as the 
distance, the other as the square of the distance of the rectangle from the fixed 
plane. 

If the rectangle has the area A when at distance H, show that the volume gen- 
erated is \AH. 

6. The axes of two equal cylinders of revolution intersect at right angles. The 
solid common to them both is called a groin. Find its volume. 

Let Ox and Oy be the axes of the two cylinders at right angles. The quarter- 
circles OA C and OBC are one fourth of 
their bases. The plane xOy cuts the 
surfaces of the cylinders in the straight 
lines AE and BE. The surfaces inter- 
sect in CME. A plane DLMN parallel 
to xOy cuts the cylinders and the vertical 
planes xOC, yOC in a square, which 
moving parallel to xOy generates one 
eighth of the groin. Let x be a side of 
this square, whose distance from O is h. 
Then x 2 = a 2 - h 2 . Hence 

-1- V = f a (a 2 - h 2 )dh - \o?. 

The volume of the groin is \ 6 a*, 
Fig. 106. where a is the radius of the cylinders. 

Knowing that any figure drawn on a 
cylinder rolls out into a plane figure, show that the entire surface of the groin 
is 1 6a 2 . 

7. Ox, Oy, Oz are three straight lines mutually at right angles to each other. 
A cylinder cuts the plane xOy in an ellipse of semi-axes OA = a, OB — b\ and 
the plane xOz in an ellipse with semi-axes OA = a, OC = c. The generating 




268 



APPLICATIONS OF INTEGRATION. 



[Ch. XXIII. 



lines of the cylinder are parallel to EC. Show that the volume of the cylinder 
hounded by the three planes xOy, yOz, zOx is \abc. 

8. A right cylinder stands on a horizontal plane with circular base. Show that 
the volume cut off by a plane through a diameter of the base and making an angle 
a with the plane of the base is |a 3 tan a. 

9. On the double ordinates of the ellipse b 2 x 2 -{- a 2 y 2 = a 2 b 2 , and in planes per- 
pendicular to that of the ellipse, isosceles triangles of vertical angle 2a are con- 
structed. Show that the volume of the solid generated by the triangle is ^ab 2 cot. a. 

10. Two wedge-shaped solids are cut from a right circular cylinder of radius a 
and altitude h, by passing two planes through a diameter of one base and touching 
the other base. Show that the remaining volume is (n — ^)a 2 h. 

11. Two cylinders of equal altitude // have a circle of radius a for their common 
base; their other bases are tangent to each other. Show that the volume common 
to the cylinders is \a 2 h. 

12. A cylinder passes through two great circles of a sphere which are at right 
angles. The volume common to the solids is {\-\-\Tt)/it times that of the sphere. 

13. Two ellipses have a common axis and 
their planes are at right angles. Find the 
volume of the solid generated by a third ellipse 
which moves with its center on the common 
axis, its plane perpendicular to that axis, and 
its vertices on the other two curves. 

Let AOC and A OB represent quadrants of 
the given ellipses. 

OA = a, OB = b, OC = c. 

Then LMN represents a quadrant of the 
moving ellipse, having z andj as semi-axes. 
Let x = OMbe the distance of the plane LMN 
from O. The area of the moving ellipse is nyz. 




Ah 



Hence we have for the volume 



c 2 x 2 + a 2 z 2 - a 2 c 2 and b 2 x 2 -f a 2 y 2 = a 2 b 2 . 



{ityz) dx — \itabc. 



The surface is called the ellipsoid with three unequal axes. 

14. Two parabolae have a common axis and vertex. Their planes are at right 
angles. Find the volume generated by 
an ellipse which moves with its center 
on the common axis, its plane perpen- 
dicular to that axis, and its vertices 
on the parabolce. 

Let OM and ON be the two parabolae 
whose equations referred to AOL, BOL 
as axes are x 2 = 2a 2 z andjj' 2 = 2b 2 z. 

MLN is the position of a quadrant 
of the generating ellipse at a distance 
z = OL from O. The area of the 
ellipse is itxy. The volume generated 
from z = o to z = c is 



V 



— I (izxy)dz = itabc 2 . 



The surface generated is called the 
elliptic paraboloid. 




Fig. ioS. 



ART. 179.] ON THE VOLUMES OF SOLIDS. 269 

15. Volume of the hyperbolatoid. 

Given two parallel planes at a distance apart //. The solid cut out between 
the planes by a straight line intersecting them and moving in such a manner as to 
return to its initial position is called the hyperbolatoid. 

If in one of the planes a fixed point P be taken, then a straight line through 
P, moving always parallel to the line generating the curved surface of the hyper- 
bolatoid. cuts out a cone between the planes, called the director cone of the hyperbo- 
latoid. Show that the volume of the hyperbolatoid between the parallel planes is 
equal to 

where B v B 2 are the areas of the sections of the solid by the parallel planes dis- 
tant apart //, and C is the area of the base of the director cone. 

Hint. Any plane parallel to the given planes cuts the generating line in seg- 
ments that are in constant ratio. Therefore the area B of any such section is 

B = k x B 2 -f k 2 B x - k x k 2 C 

(projecting on a plane parallel to the bases), by Elliott's theorem, § 164, (3). 

k\ and k 2 can be expressed in terms of A, the distance of the section B from 
either base B x or B r Then 

V= / Bdh, 

where B is a quadratic function of h, and the result follows directly. 

Since i?is a quadratic function of h, the results of Exercises 10, 11, 16, Chapter 
XX, apply also to the hyperbolatoid, when ordinates are read sectional areas. 

An important general case is : If the generating straight line moves in such a 
manner as to remain always parallel to a fixed plane, then C = o and 

16. Find the section of minimum area in a given hyperbolatoid, and show that 
sections equidistant from the least section have equal areas. 

17. On the double ordinate of x 1 -f- y 1 = a 2 , as a central diagonal, is con- 
structed a regular polygon of n (even) sides, whose plane is perpendicular to that 
of the circle. Show that the volume generated by the polygon is 

. 27T 

sin — 

3 27t » 

n 
and therefore the volume of the sphere is f 7Ta 3 . 

18. Show that the hyperbolic paraboloid passing through any skew quadrilateral 
divides the tetrahedron having for vertices the corners of the quadrilateral into two 
parts of equal volume. 

19. On a sphere of radius R draw two circles whose planes are parallel and 
distant R/ \Z$ from the center of the sphere. Draw tangent planes to the sphere 
at the ends of the diameter perpendicular to the planes of the circles. 

Show that any ruled surface passing through the circles cuts out a solid between 
the tangent planes whose volume is equal to that of the sphere. 



BOOK II. 

FUNCTIONS OF MORE THAN ONE 
VARIABLE. 



271 



PART V. 

PRINCIPLES AND THEORY OF DIFFERENTIATION 
CHAPTER XXIV. 



THE FUNCTION OF TWO VARIABLES. 

180. Definition. — When there is a variable z related to two other 
variables x and y in such a manner that corresponding to each pair 
of values of x, y there is a determinate value of z, then z is said to 
be a function of the variables x and^y. 

We represent functions of two variables x, y by the symbols 
f(x, v), <p(x, y), etc., in the same sense that we employed the corre- 
sponding symbols f{x) } <p{x), etc., to represent functions of one 
variable x. 

When it is so well understood that we are considering a function 
/(x, y) of the two variables x and y that it is unnecessary to place 
the variables in evidence, we frequently omit the variables and the 
parenthesis and represent the function by the abbreviated symbol f. 
In like manner we frequently consider the single letter z as represent- 
ing a function of the variables x and^, and write 

181. Geometrical Representation. — Let z be a function of two variables x 
and y. Let the value c of z correspond to the values a of x and b oiy. Through 
a point O in space draw three straight lines Z 

Ox, Oy, Oz mutually at right angles, in such a 
manner that Oz is vertical as in the figure. We 
then have a system of three planes xOy, yOz, 
zOx mutually at right angles, of which xOy is 
horizontal. These planes divide space into eight 
octants. The plane xOy we take as the plane 
of the variables x andjj/, in which we represent 
any pair of values of the variables x and y by a 
point having these values as coordinates referred 
respectively to Ox, Oy as axes, as in plane ana- 
lytical geometry. 

We take, as in the figure, Ox drawn to the 
right as positive, drawn to the left as negative; Oy drawn in front of the xOz plane 
as positive, drawn behind that plane as negative; Oz drawn upward above the hori- 
zontal plane as positive, drawn downward as negative. 

273 




274 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIV. 

To represent the value z = c of the function corresponding to the values x = a, 
y — b of the variables : Construct the point A^in the plane, xOy, of the variables, 
having for its coordinates OM = a, MN = b. The value c of the function z can 
then always be represented by a point P, which is constructed by drawing a per- 
pendicular NP to the plane of the variables at A r , such that NP — c is drawn 
upwards or downwards according as c is positive or negative. 

The representation is nothing more than the Cartesian system of coordinates in 
analytical geometry. The numbers a, b, c, or in general x, y, z, are the coor- 
dinates of the point P with respect to the orthogonal coordinate planes xOy, vOz, 
zOx. 

We can then always represent any determinate function f(x, y)of two variables 
by a point in space whose distance from a plane is the value of the function. 

182. Function of Independent Variables. — Let z = f(x,y) be a 

function of the two variables x and y. When there is no connection 
whatever between x and^', then z is said to be a function of the two 
independent variables x and r. 

This means that, within the limits for which z is a function of x 
and y, whatever be the arbitrarily assigned values of x and^' there 
corresponds a value of z. 



Geometrical Illustration. 
Consider the function of two independent variables 



+ \/a 2 - x l - ?K 

This function has no real existence for values of x and y such that x 2 -\-y 7 > a 2 . 
Also, for x' 1 -f- y 2 = a 2 the function is o, while for any arbitrarily assigned values 
of x and/ whatever, such that x 2 ~\-y 2 < a 2 , the func- 
tion has a unique determinate positive value. Geo- 
metrically speaking, the function exists for any point 
on or inside the circumference of the circle x 2 -\-y 2 = a 2 
in the plane xOy, and the point representing the 
function for any such assigned pair of values of x, y, 
is a point on the surface of the hemisphere 

x 2 +y 2 + z 2 = a 2 

which lies above xOy. The circle x 2 -\- y 2 = a 2 is 
called the boundary of the region of the variables for 
which the function 




z = + \/a 2 - x 2 - y 2 

is defined, or exists in real numbers. 

In general, a function z = f(x, y) of two independent variables is represented 
by the ordinate to a surface of which z = f(x, y) is the equation in Cartesian 
orthogonal coordinates. The study of a function of two independent variables 
corresponds, therefore, to the study of surfaces in geometry, in the same sense that 
the study of a function of one variable corresponds to the study of plane curves as 
exhibited in Book I. 

183. Function of Dependent Variables. — Let z = f(x,y) be a 
function of two independent variables x and y. Since x and y are 
independent of each other, we can assign to them any values we choose 
in the region for which z is a denned function of .v and_>>. 



Art. 183.] THE FUNCTION OF TWO VARIABLES. 



I. In particular, we can hold y fixed an 
which case z is a function of the single 
variable x. For example, let y r= b be 
constant, then 

Z =/(.V, i) (I) 

is a function of the single variable x. 
If z =zf[x,y) be represented by a sur- 
face, then equation (1), which is nothing 
more than the two simultaneous equa- 
tions 

z=/(x,y), 
y=b, 

is represented by a curve AB in a plane x'O'z' , parallel to and at a 
distance b from the coordinate plane xOz. Or, is the curve of inter- 
section of the surface z =zf(x,y) and the vertical plane y — b, as 
exhibited by the simultaneous equations (2). The equation z =f(x, b) 
of this curve is referred to axes O'x' ', O'z' of x and z respectively, in 
its plane x'Oz\ 



(*) 




II. In like manner, if 




we make x remain constant, say x = a, 
and let_>> vary, then z = f(x 9 y) becomes 

*=A*,y)> (3) 

a function of y only, and is represented 
by a curve AB in a plane y'O'z', Fig. 
112, parallel to and at a distance a from 
the coordinate plane yOz. Or, it is the 
curve of intersection of the surface 
z =f(x,y) and the plane x = a, whose 
equations are 

z=/(x,y), J 

x = a. \ 



(4) 



Fig. 112. 



III. Again, since x and y are inde- 
pendent, we can assign any relation we 
choose between them. For example, instead of making, as in I, II, 
x and y take the values of coordinates of points on the line x = a 
or y = b in xOy, we can make them take the values of coordinates 
of points on the straight line 

x — a y — b 

= -■ = r > (5) 

cos a sin a 

which is a straight line through the point a, b in xOy and making 
an angle a with the axis Ox. 
Substituting 

x = a -f- r cos a, y = b -{- r sin a 



276 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIV. 

in z =/[x,y) for x and y respectively, and observing that r is the 

distance of x, y from a, b measured 
• on the line (5), we have 




=/(< 



r cos i\ 



sin a). (6) 



Fig. 113. 



If a is constant, (6) is a function of 
the single variable r, and is the equa- 
tion of a curve APB cut out of the 
surface z ■=/{x i y) by a vertical plane 
through (5), and the curve has for its 
equations 



a y — b 



(7) 



cos a 



sin a 



The curve (6) is referred in its own plane, rO'z\ to O'r, O'z' as 
coordinate axis. The coordinates of any point P on the curve being 




IV. In general, x and _y being independent, we can assume any 
relation between them we choose. 

For example, we may require the 
point x, y in xOy to lie on the curve 

<p(x,y) = o. 

Then, as in III, z = f(x, y) is a 
function of the dependent variables x and 
y which are connected by the functional 
relation (p(x, y) =. o. The geometrical 
meaning of this is: The point P repre- 
senting the function z must lie in the 
vertical through the point P' represent- 
ing x, y on the curve <p(x,y) = o. Or, 

the function z of the dependent variables x } y is represented by the 
ordinate to a curve in space drawn on the vertical cylinder which has 
the curve A'P' for its base. The curve A ' P ', whose equation myOx 
is 0(a-, y) = o, is the horizontal projection of the curve in space AP 
representing the function. 

Geometrically speaking, the function z = f(x, y) of two depend- 
ent variables x and r, connected by the relation cp(.\\ y) = o, is 
represented by the space curve which is the intersection of the surface 
z-—/[x i y) and the vertical cylinder 0(.v, r) = o, whose equations 
are 

*=A*.J>). \ 

= <p{x,y). I 



(8) 



Art. 1S5.] 



THE FUNCTION OF TWO VARIABLES. 



77 



say 

(0 

the 



If we solve 0(.v, y) = o for y and get y — tj-(x), then substitut- 
ing for y \nf{x,y), we express as a function of x only, thus : 

C =/[.V, »/•(.!■)]. ( 9 ) 

This equation (9) is the equation of the projection of the space 
curve AP (8) on the plane zOx. 

In like manner we can express z as a function of r only, and get 
the equation of the orthogonal projection of (8) on the plane vOz. 

184. The Implicit Function.— We saw in Book I how the 
functional dependence of one variable on another was expressed by 
the implicit functional relation, or equation in two variables, 

fix, y) = o, 

and that this implied or defined either variable as a function of the 
other. We also saw that this functional relation could be repre- 
sented by a plane curve having x and y as coordinates of its points. 
The implicit function of two variables is a particular case of a func- 
tion of two independent variables. For, in such a function, 

z=/{x,y), 

of the two independent variables x and y, if we make z constant, 
z = c, we have the implicit function in two variables 

A*> y) = c - 

Geometrically, this is nothing more than the equation to 
curve LMN, Fig. 115, cut out of the 
surface z = /[x, y) by the horizontal 
plane z = c, at a distance c from xOy. 

Its equations are 

*=A*>J>)> \ 

Z = C. ) 

The lines cut on a surface by a 
series of horizontal planes are called 
the contour lines of the surface. In 
particular, if z = o, then f{x, y) = o 
is the equation in the xOy plane of the horizontal trace of the surface 
2 = f(x, y), or the curve ABC cut in the horizontal plane by the 
surface. 

In the same way that the implicit equation in two variables 
defines either variable as a function of the other, the implicit function 

/(jv, y, z) = o 

is an equation defining either of the three variables as a function of the 
other two as independent variables, and can be represented by a 
surface in space having x, y, z as the coordinates of its points. 

185. Observations on Functions of Several Variables, — The 

general method of investigating a function of two independent 
variables is to make one of the variables constant and then study the 



(-) 




Fig. 



5- 



278 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cll. XXIV. 

function as a function of one variable. Geometrically, this amounts 
to studying the surface represented by investigating the curve cut 
from the surface by a vertical plane parallel to one of the coordinate 
planes. 

Or, more generally, to impose a linear relation between the 
variables x and y, and thus reduce the function to a function of one 
variable, as in § 183, III, which can be investigated by the methods 
of Book I. Geometrically, this amounts to cutting the surface by 
any vertical plane and studying the curve of section. 

As we have seen in § 184, and as we shall see further presently, 
the study of functions of two variables is facilitated by reducing them 
to functions of one variable, and reciprocally we shall find that the 
study of functions of two or more variables throws much light on the 
study of functions of one variable. 

186. Continuity of a Function of Two Independent Variables. 
Definition. — The function z = f{x, y) is said to be continuous at 
any pair of values x, y of the variables when corresponding to x, y 
we have/~(..r, y) determinate and 

for x x (=)x, y x (^=.)y J independent of the manner in which x x and y x are 
made to converge to their respective limits x andy. 
The definition also asserts that 

£[f(x 1 >y 1 )—A*>j>)]^.°> 

tor x x (=)x, y x (=)v. 

In words: The function z =f(x,y) is continuous at x,y when- 
ever the number z x = f(x\,y l ) converges to z as 
a limit, when the variables x x , y x converge 
simultaneously to the respective limits x,yin 
an arbitrary manner. 

Geometrically interpreted, the point P xi 
representing x\ , y x , z x , must converge to P, 
representing x, y, z, as a limit, at the same time 
that the point N, representing x x , j\ , con- 
verges to M, representing x,y; whatever be the 
path which N is made to trace in xOy as it 
converges to its limit M. 
A function _/"(.#, y) is said to be continuous in a certain region A 

in the plane xOy when it is continuous at every point x, y in the 

region A. 

An important corollary to the definition of continuity oi/(x y y) 

at x, y is this: Whatever be the value of f(x, y) different from o, we 

can' always take .v , y so near their respective limits x, y that we 

shall have/i^Vj , y x ) of the same sign a.sf(x, y). 

187. The Functional Neighborhood. — A consequence of the 
definition of continuity of z =zf(x,y) is as follows: 




Art. 187.] 



THE FUNCTION OF TWO VARIABLES. 



2 79 



If /(.i\ y) is continuous in a certain region containing </, b, we can 
always assign an absolute number e so small that corresponding to 
e there are two assigned absolute numbers h and /-, such that for all 
values of x and y for which 

i*-*i < K \y — *\ <k, 

we have 

I/KjO-./M)| <e. 

The proof of this is the same as that given for a function of one 
variable. For, \ety and a be fixed numbers, and let x vary. Then 
whatever number \e be assigned, we can always assign a correspond- 
ing number h 9^ o, such that for \x — a\ < ^ we have 

I/O',;') -/(«,;■) I <H 

since f{x, y) is a continuous function of one variable jf, and its limit 
isf(a,y). 

In like manner for \y — b\ < k we have 



and on addition 



for all values of x, 



\A*,y)-A°,i)\ < 

such that 



x — a 



Geometrically speaking, whatever be the 
can always assign an arbitrarily 
small number e, corresponding to 
which there is a rectangle KLMN 
in the plane xOy, the coordinates 
of whose corners are K, (a — h, 
b + k); L, (a + h, b + k); M, 
(a + k, b - k); N, [a — h, b — k), 
such that, whatever be the point 
x, y in the rectangle KLMN, the 
corresponding point x, y, z on the 
surface z =/(x,y) lies between the 
parallel planes z = c-~ e, (STUV) 
and z = c + e, ( WXYZ). The 
point P representing a, b, c. 

Such a region KLMN is called 
the neighborhood of the point a, b. The point is called its center. In 
like manner the corresponding parallelopiped STUV-WXYZ is 
called the neighborhood of the point P in space. 

The above results may be stated thus: When the variables x, y 
are in the neighborhood of a, b, then must the continuous function 
f{x, y) be in the neighborhood of f\a t b). 

An important consequence is this: li/(x, y) is continuous in the 




2 So PRINCIPLES AND THEORY OE DIFFERENTIATION. [Ch. XXIV. 

neighborhood of /{a, b) 9^ o, then we can always assign a neighbor- 
hood of a, b such that for all values of x, y in this neighborhood 
the value _/(jr, y) of the function has the same sign as /{a, b). 

EXERCISES. 

1. Trace the surface representing the function 

f(x, y) ee y — mx -|- b. 

Put z = y — mx -\- b. When z = o, the surface cuts xOy in the straight line 
v = mx — b. If x = a, we have for the section of the surface by the plane x = a 
the straight line 

z = y — ma -\- b. 

Whatever be a, this line is sloped 45 ° to the plane xOy. As x = a varies, this 
line moves parallel to itself, intersecting the fixed line y = mx — b in xOy, and 
therefore generates a plane. 

In like manner it can be shown that the implicit function of the first degree in 

f(x, y, z) = Ax + By+Cz + D = o, 

is always represented by a plane. 

2. Show that the function 

f/ a 8 _ x * - yl 

can be represented by a sphere, by showing that it can be generated by a circle 
whose diameters are the parallel chords of a fixed circle, and whose planes are per- 
pendicular to that of the fixed circle. 

3. Trace the surfaces representing the implicit functions 

x 1 y 2 z 2 x 2 y 2 

-5 + l3±-2— I= =°. -i^-22^0 

a 2 o 2 c 2 a b 

by their plane sections. 

4. Trace by sections the surface representing 

(x 2 - az)\a 2 - x 2 ) - x*y 2 - o. 

5. Find the maximum value of the function 

x 2 v 2 

when the variables are subject to the condition x -j- y = 1. 

Let z =/(x, y). Then z is immediately reduced to a function of one variable 
by substituting I — xioxy. 

' " a- b 2 ' 

dz 2X (I - x) 

••• jx-=-^+ 2 —J^-=° 

gives x = a 2 /(a 2 + ^). y = b 2 /(a 2 + b 2 ), z = 1 - i/(a 2 + b 2 ), which is a 
maximum value of z since D%z is negative. 

Consider the geometrical aspect of this problem. We have 

. _ x* )' 2 

Z ~ X "a 2 ~~J 2 ' (I) 

the equation of the elliptic paraboloid whose vertex is O, o, 1, and which cuts xOy 
in the ellipse x 2 /a 2 -\- y 2 /b 2 = 1. 



Art. 1S7.J 



THE FUNCTION OF TWO VARIABLES. 



We wish the highest point on the curve cut out of the surface by the plane 
x-\-y = I. Take OV, the horizontal trace of this plane, as the positive axis 
of r, and O'z', its vertical trace on yOz, as axis 
of z in the plane rCz'. Then for the equation 
to the curve in the plane x -\- y = 1, or 

x — o y — I 




Fig 



y 2 — 4/2 

we substitute x = r 4/2, y = 1 — r 4/2 in (1). 
Hence the equation to the curve of section in its 
own plane is 

/ i\ , 2 4/2 /a 2 4- /A , 

Z> r c = o gives r — a 2 /(a 2 -f- ^ 2 ) 4/2", and 
Z>|r = — . Hence the values of x, y, z as before. 

The first method, in which we substitute for/ in terms of x, is only possible 
when we can solve the condition to which the variables are subject, with respect to 
one of them. The second method, in which we express x and y in terms of a third 
variable, is always possible, although perhaps cumbersome. 

The class of problems such as the one proposed and solved here should be care- 
fully considered, for we propose to develop more powerful methods for attacking 
them. But it should not be forgotten that those methods themselves are developed 
in the same way as is the solution of this particular problem. The student should 
accustom himself to seeing curves referred to coordinate systems in other planes 
than the coordinate planes, for in this way a visual intuition of the meaning of the 
change of variables, and a concrete conception of the corresponding analytical 
changes which the functions undergo, is acquired. 



CHAPTER XXV. 
PARTIAL DIFFERENTIATION OF A FUNCTION OF TWO VARIABLES. 

1 88. On the Differentiation of a Function of Two Variables. — 

A function of two independent variables has no determinate deriva- 
tive. It is only when the variables are dependent on each other that 
we can speak of the derivative of a function of two variables. The 
derivative of a function of two variables is indeterminate unless the 
variable is mentioned with respect to which the differentiation is 
performed and the law of connectivity of the variables given. 

189. The Partial Derivatives of a Function of Two Independ- 
ent Variables. — Among all the derivatives a function of two variables 
can have, the simplest and most important are the partial derivatives. 

Let z = f(x,y) be a function of the two independent variables 
x and y. The simplest relation we can impose between x and y is 
to make one of them remain constant while the other varies. We 
then reduce the function 2to a function of one variable, to which 
we can apply all the methods of Book I for functions of one variable. 

For example, \ety be constant and x variable. Then z =f(x, y) 
is a function of x only, and it can be differentiated with respect to 
x by the ordinary method, and we have 

D ±Ax x >y) -f(x,y) 



1- 



This is called the partial derivative of the function z or f with 
respect to x. To obtain the partial derivative of f(x, y) with respect 
to x, makejy constant and differentiate with respect to .v. 

Correspondingly, the partial differential of /[x, y) with respect to 
x is the product of the partial derivative with respect to x, D x f, and 
the differential of x or x\ — x = Ax. If we represent the partial 
differential of/* with respect to x by d x f, then we have 

d x f = f x (x,y)dx, 

and the corresponding partial differential quotient is 

It is customary to employ the peculiar symbolism designed by 

282 



Art. 190.] 



THE FUNCTION OF TWO VARIABLES. 



2»3 



Jacobi for representing the partial differential quotient or derivative 
of /{x, y) with respect to x. Thus the above will hereafter be 
written (the svmbol 3 is called the round d) 

df = d f f 

dx ~ dx ' 

The symbol d being used instead of d to indicate the partial 
differential as distinguished from what will presently be defined as 
the total differential, which will be represented as formerly by d. 

In the same way, if we make x cons/an/, then f[x, y) becomes a 
function of one variable y, and has a determinate derivative with 
respect toy. This derivative we call the partial derivative oi/(x,y) 
with respect toy, which is written and defined to be 



dy 






>i( = )> 



A x >yi) -A x >y) 
y x -y 



190. Geometrical Illustration of Partial Derivatives. — If 
z =f{x,y) is represented by the ordinate to a surface, then at any 
point P(x, y, z) on the surface 
draw two planes PMQ and PMR 
parallel respectively to the coor- 
dinate planes xOz and yOz. These 
planes cut out of the surface the 
two curves, PK und PJ respectively, 
passing through P. 
z=f(x,y) (y constant) 

is the equation of the curve PK in 
the plane PMQ. 
z=f(x,y) (x constant) 

is the equation of the curve P/in the plane PMR. 

Draw the tangents PT 'and PS to the curves PK and P/in their 
respective planes, and let them make angles and ip with their 
horizontal axes, as in plane geometry. Then we have 




— = tan 0, 
ox 



dz 
3y 



tan xp. 



Therefore the partial derivatives of f(x, y) with respect to x and 
y are represented by the slopes of the tangent lines to the surface 
z =f(x, y), at the point x,y, z, to the horizontal plane xOy. These 
tangents being drawn respectively parallel to the vertical coordinate 
planes xOz, yOz. 

Also, draw PV parallel to MQ, and PU parallel to MR. Then 
we have 

VT = (x l - x) tan 0, US = (y f -y) tan fa 



284 PRINCIPLES AND THEORY OF DIFFERENTIATION. [C11. XXV- 
if Q is x lt y, and A' is x, y' . Or 

represent the corresponding partial differentials of f with respect to 
x and^' at /*(.*, y, z). 

Thus the partial derivatives and differentials of f(x i y) are 
interpreted directly through the corresponding interpretations as 
given for a function of one variable. 

191. Successive Partial Derivatives. — If z = /[x, y) is a func- 
tion of two independent variables x and y, then, in general, its 
partial derivative with respect to x, 

is also a function of x and y as independent variables. This deriva- 
tive can also be differentiated partially with respect to either x or y, 
as was /(or, y). Thus, differentiating again with respect to x, y being 
constant, we have the second partial derivative of f with respect to x. 
In symbols 

d 2 /(x,y) _ 

dx* -/"(*' J')- 

In like manner/*^r, y) can be differentiated partially with respect 
tor, a- being constant. Thus we have for the second partial differen- 
tial quotient of_/~with respect first to x and then toy 

dydx S^X'Sh 

Similarly, differentiating fy(x, y) partially with respect to_>', we 
have 

dy* ~ dy -Jyy\ x >y)> 

and with respect to x we have 

dY(x,y)_ d/;(x,y) =z 
dxdy dx Jy*y~>J)- 

Thus we see that the function z — f{y\ y) has two first partial 
derivatives, 

dz dz 

a? a7' 

and four second partial derivatives, 

d*z dh dh &z 



dx*' df' dydx 9 dxdy 
Each of these give rise to two partial derivatives of the third 



Art. 192.] 



THE FUNCTION OF TWO VARIABLES. 



28S 



order, and generally the function has 2* partial derivatives of the 
«th order, of the forms 

d n z b n z 



dx'dy*' dy>dx*' 
where p and q are any positive integers satisfying/ -(- q = n. These 
«th derivatives, however, are not all different, for we shall demon- 
strate presently that dx p and dy in the denominators are interchange- 
able when the partial derivatives are continuous functions, and that 

b n z d n z 

dxt dyi ~ dydx*' 
or the order of effecting the partial differentiations is indifferent 
The number of partial derivatives otf(x,y) of order n is then n -\- 1 

EXAMPLES. 

1. If z = x 1 -f- axy -\- cos x sin y, 

dz 
.'. ~ = 2x -\- ay — sin x sin/, 
ox 

dz 

by- 



= ax -f- cos x cos y. 



2. if f(*,y) = - 2 + 



i>- 



1, 

dx 



IX 

~a* 



3. In Ex. I, 



d 2 z 



dy~ 
dh 



-— — = a — sin xcosj/=-— — , 
q> ojc ox qj/ 



= 2 



dx 2 
4. In Ex. 2, show that 



cos x smy, 

ay 



dH 



= — cos x smy. 



ey 



dy dx dx dy 

192. Theorem. — The partial derivatives are independent of the 
order in which the operations are effected with respect to x and y. 
In symbols, if z — f(x,y), we have 
d 2 z d 2 z 



dx dy dy dx' 
Consider the rectangle of the 
four points 

M, (x,y); M v (x v y x ); 
Q, (x x ,y); R, (x,yj. 
The theorem of mean value applied 
to a function of one variable x gives 

=A'{S,y), (1) 

where B> is some number between x x and x. (See Book I, § 62.) 




286 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXV. 
Form the difference quotient of (i) with respect to_>', 

a_ 4/ /(. Wl ) -Aw) -A*,*) +A*.y) 

Ay Ax O-i-J-K^-*) 

= mS ' y ]Zf S ' y) =fW,v), (-) 

where rj is some number between j^ andj>. The value (2) is therefore 
equal to the second partial derivative of/*, taken first with respect to 
x, then with respect to y, at a pair of values £;, rj ofx,y. Geomet- 
rically, at a point B,, 77 in the rectangle MQM X R. 

In like manner, taking the difference-quotient of/" first with respect 
to y, we have 

%- *"'}-?■* -r*.«. (3) 

where rf is some number between y x and y. 

Now taking the difference-quotient of (3) with respect to x, we 
have 

A Af = /(. Wl ) -f{x,y x ) -f(x v y) +f(x,y) 
Ax Ay " (x 1 -x)(y l -y) 

— x —x ~~ - /r?/ ' * ^ ' 7 '' ^ 

where £,' lies between x x and ~r, 7;' between y Y and y. The value of 
(4) is then equal to the second partial derivative of/", taken first with 
respect to y and then with respect to x at some point B,' ', rf , also 
inside the rectangle MQM X R. 

But (2) and (4) are identically equal. Hence we have 

ayis, y) __ dy(Z>, V ') 

This relation is true whatever be the values x v y v 
If now the functions 

*S and */ 



dy dx dx dy 

are continuous functions of x and_y in the neighborhood of x,y, then 
since £,' ', rf and £, 7; converge to the respective limits x, y when 
x l ( = )x,y l ( = ]}>, the two members of (5) converge to a common limit 
at the same time, and therefore 

ay = &/ (6) 

dy dx dx dy' * ' 



Art. 192.] THE FUNCTION OF TWO VARIABLES. 287 

Incidentally, equations (2) and (4) show that the difference- 
quotients 

A Af _ AY A A/ _ AY 

Ay Ax = Ay Ax ' Ax Ay ~ Zix Ay 

converge to a common limit whatever be the manner in which 
Ax( = )o, Ay( = )o, and that common limit is 

by dy 

or 



dy dx dx dy ' 

Observe that in the symbols 

dydx J ~ Jxy 

the operations are performed in the order of the proximity of the vari- 
able to the function. 

In like manner, making use of the result in (6), we have 

dy\_ d d d/ _ d d df d dy 
dy) dx dy dx dy dx dx ~~ dy dx 1 ' 

dy dy 

dx 2 dy dy dx 2 ' 
and similarly for other cases. Hence, in general, 

tf+Qf & + <!/ 

dx* dy* ~~ dy q dx p ' 
in whatever order the differentiations be made. 



!L(3 

dx \dx 



EXERCISES. 



x 
1. If z — tan- 1 - , show that 



d-: 



dx dy dy dx (x 2 -j- y 2 f 

2. If z = -^— 2 , find D% y z, D; x z. 

3. Verify in the following functions the equation 

ay ay 

dx dy ~ dy dx' 
x sin y -f- y sin x, log tan (y/x), 

x logy, (ay - bx)/(by - ax), 

xy, y log (1 -\-xy). 



4. If 


z — tan— 1 , show that 




d 2 z 1 d 4 z i$xy 


5. If 


dx dy ~ (I + X 2 _|_ yz)i ' dx 2 dy 2 ~ ( X _|_ x i _|_ y if 

u = x z y 2 — 2 xy 1 -|- 2 x2 }' A i show that 
du du 



288 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXV. 



6. If t — log (e* -f- eP), then 



~da 



dt 



= i. 



7. If c(e* + e b ) 



€**>, then 



dc dc 



8. If z — e x sin y -\- ey sin x, show that 
\ 2 /S^X 2 

7 + W = '" + ^ + 2eX+y Sin {X +y) ' 



/dz' 



9. From z = xvy*. show that 



9^ dz 



(■* + y + lo g *)* 



10. Show that if ^ is the angle between the plane xOy and the tangent plane to 

the surface z = f(x, y) at x. y, then 

— (©■+©•. 

Let /*. (x, / , z) be on the surface, and 
PRS the tangent plane. 

Draw MN — / perpendicular to RS. 
Then ^ = iM". 

9/ _ .P/I/ 9/ _ PM 
dJc ~ RM' dy ~ SM' 

Since RS. NM - RM- MS, 
and RS 2 = RM 2 + JfS*, 

.-. (MN)~* = (RM)~* + (MS)**, 
and therefore 




tan 2 ;' = 



MAT* 



I PM\* 
\~RM 



Also, 



)'+ (m) - ff ) + (I) 

(I) 2 - (!) '■ 



11. In Fig. 121, let A 7 " be any point in the trace of the tangent plane with xOy. 
Let NM make an angle 9 with (9x, and the tangent line NP to the surface make an 
angle </>with the horizontal plane xOy. Then the triangle RSM is the sum of the 
triangles RMJV, NMS, or 

RM-SM = SM-NM cos 6 + RM-NM sin 0, 

PM PM „ , iW . _ 
•A^- = -^7 cosG + -^7 sm0 - 



Therefore the slope of a tangent line to the surface at x, y, z, whose vertical 
plane makes the angle with zOx, is 

^_ C os + -/ - 

9x ~ 9) 



tan 



(0 



12. Find the tangent line to a surface at /'which has the steepest slope. 
From Ex. II we have 

— tan = — — sin -f i- cos G = o. 
,/Q ^ dx dy 



Art. 192.] 



THE FUNCTION OF TWO VARIABLES. 



89 



The values of sin S, cos from this equation put in (1), Ex. II, give for the 
tangent line of steepest slope 

-*-© + ©■ 

Observe that this is the slope of the tangent plane in Ex. 10. 
13. If <p(x. y) = o is the equation of any plane curve, show that 

dy 



dx 



dx 

d<P(*, y) ' 

dy 




Also, 



tan MPQ 



tan MPQ 
~ tan NP X Q' 
d(p(x v 7]) 



Let z = <p(x, y) be the equation of a sur- 
face cutting the horizontal plane in the curve 
4>(x, y) = o. 

Let P, (x,y) and P x , (x„ y x ) be two '2/ 
points on the curve <p{x, y) — o. Draw the 
vertical planes through P and P x parallel 
respectively to xOz and yOz, cutting the sur- 
face in curves PQ, P X Q. Then Q is a point 
x v )>■> z on tne surface. The derivative of y 
with respect to x in (p{x, y) = o is the limit of the difference-quotient 

y x _ y _ MP X _ MQ cot MP X Q __ tan MPQ 
x x ~^x ~ PM ~ MQ cot MPQ ~ tan MP X Q 

^ being between x and ^ 1? 77 between j and / x (by the theorem of the mean). 
Therefore, when x x ( = )x, y x (=)y, 

d(p 

d y_ _ £ y x - y _ _ ^^ 

dx T x x — x d(p' 

~¥ 

This usually saves much labor in computing the derivatives of implicit functions 
in x and y. 

The important results of Exs. 10, II, 12, and 13 are deduced here geometrically 
to serve as illustrations of the usefulness of partial differentiation. They will be 
given rigorous analytical treatment later. 

14. Employ the methods of Book I, and also that of Ex. 13, to find D x y in the 
following curves: 

x^/a 1 — y 2 /b 2 —1=0, x sin y — y sin x = o, 

ax % y -j- by 2 x — \xy = o, e x sin y — log y cos x = o. 

15. Show that the slope of the tangent at x, y on the conic 

ax 2 -f- by 2 -f- 2hxy -)- 2ttx -\- 2vy -f- d = o 
dy ax -f- hy -f- u 

lS 1x ~ ~~ hx -\- by + v 



CHAPTER XXVI. 
TOTAL DIFFERENTIATION. 

193. In the partial differentiation of f{x, y) we made x or y 
remain constant during the operation, and differentiated the function 
of the one remaining variable by the ordinary methods of Book I. 

We now come to consider the differentiation ofy^, y) when both 
x and y vary during the operation of evaluating the derivative. Such 
derivatives are called total derivatives. 

In order to make clear the nature of the total derivative of a 
function 

z=f{x,y), 

consider the simple case when there is a linear relation between x 

\nd i', 

x — x' y — y' _ 

7 = ~lnr " r ' 

where / = cos 6, m = sin 6, and the differentiation is performed with 
respect to r. Let x', y'\ I, m ; be constant. Then r varies with x 
and y, and 

r*=(x- x'f -f- (y - y')\ 

Also, x andjy are linear functions of r, and 

x = x' -J- Zr, y =y f -|- mr. 

Substituting these values of x and y in f(x, y), we reduce that 
function to a function of the one variable r, and it becomes 

f( x > + lr,y' + mr). (1) 

The derivatives of this function with respect to r can now be 
formed by the methods of Book I. Thus we get by the ordinary 
process of differentiation 

W ?L W etc 
dr' dr^ dr*' " 

for the successive derivatives of f with respect to r. These are 
called the total derivatives of/* with respect to r. Both variables x 
and r vary with r. 

We can give a geometrical interpretation to this total derivative 
as follows: The equation 

x — x> y —y' r , v 

— j — — — r ( 2 ) 

l m v ' 

290 



Art. 194.] 



T< )TAL PIFFEREXTIATK )N. 



291 



is the equation of a straight line through x', y' in the horizontal 
plane xOv, making an angle 6 with Ox. r being the distance 
between the points x', y' and x, y on the line. Let 0' be the point 
j/f.y. Draw O'z' vertical. The vertical plane rO'z' through the 




Fig. 123. 

line (2) cuts the surface representing z=f(x,y) in a curve PP X , 
whose equation in its plane, referred to O'r and O'z' as axes of 
coordinates r and z, is 

z=f{x' + lr,y' + mr). (3) 

Let P 1 be a point on this curve whose coordinates in space are 
x x , y\ , z x and in rO'z' are r x , z v Let r x — r = Ar. Then, by 
definition, the derivative of z with respect to r at x, y is the limit 
of the difference- quotient, when r 1 ( = )r, 

*, - *_A x i>yi) -A x >y) 







r i~ 


- r 




r , ~ 


- r 


Hence 


we 


have 




dz _ 
~dr ' ~ 


= tan 


GO, 



where go is the angle which the tangent PM" to the curve PP X at 
P, and therefore to the surface, makes with O'r, or the horizontal 
plane xOy. 

Observe that as a? , y converge to x, y, the point M converges 
to M along the line M X M. 

By assigning different values to 6 we can get the slope of any 
tangent line to the surface, at P, with the horizontal plane. 

In particular, when the line (2) is parallel to Ox or Oy, or, what 
is the same thing, when 6 = n or \n, the total derivative becomes 
a partial derivative, as considered in the preceding chapter. 

194. The Total Derivative in Terms of Partial Derivatives. — 

It is in general tedious to obtain the total derivative, after the 
manner indicated in § 193, by reducing the function directly to a 



292 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVI. 

function of one variable, and generally it is impracticable. We now 
develop a method of determining the total derivative in terms of the 
partial derivatives. Let z =f(x, y), where x and y are connected 
by any relation cf>(x, y) — o. To find the derivative of z with respect 
to /, where / is any differentiate function of x and y. 

Let z take the value z lt and / become / 1? when x, y become 

Let v be constant and x 1 be a variable. Then the law of the 
mean is applicable to the function f(x y , y) of the one variable x\, 
and we have 

A*, <y) -A*>y) = (•*, - *)^AZ>y). (1) 

where £ is some number between x l and x. 

In like manner, let x be constant and_>> vary, then, by the law of 
the mean, 

/(. Wl ) -A^.y) = 0\ -yy^A*i> n)> ( 2 ) 

where ?; is some number between y x and y. 
Adding (1) and (2), we have 

A*> , >\) -/(•*, >') = K - x )d$A5>y) + O'i -y) rrj A*i >v)- (3) 

Therefore the difference-quotient with respect to / is 

z j^r t =A{S,y)^+A{^v)% (4) 

Ax, Ay, Az, At converge to o together, and at the same time 
x x (=)x, ^(=)^,j^ 1 (=)y, V( = ]>'- Also > 

¥(Z>j>) and *A*i>v) 



dg drj 

have the respective limits 

y&y) and sa^ 

ox oy 

if these latter functions are continuous in the neighborhood of x, y. 
Passing to the limit in (4), we have for the total derivative oif(x,y) 
with respect to /, at x, y, 

df^dfdx dfdy 

dt dx dt "*" dy dt' ^ 5) 

The geometrical interpretation of (1) is this: In Fig. 123 we 
have M t (x,y); M %i [x lf yfr (J, (-x\,y); R, (x, yj. 
Also, 

A*x>y) -A x >y) = Q' K = p Q' tan qtk. 



Art. 195.] TOTAL DIFFERENTIATION. 293 

But, since on the curve PK there must be a point A", (4f, y, z) at 
which the tangent is parallel to the chord, 

In like manner for equation (2), 

AWi) -A*vJ) =£P X = - ^A'tan LKP V 
But, since there is a point F, (x x , t], z) on the curve KP X at which 
the tangent is parallel to the chord, we have 

- tan ZA^ 1 = 1/(^,7,). 

195. The Linear Derivative. — An important particular total 
derivative is the case considered in § 193. Suppose there is a linear 
relation between x and^', such as 

x — a y — b 



I m 



= r. 



Then x = a -f- Ir, y = b -J- mr. To find the total derivative of 
f(x, y) with respect to the variable r, we have 

dx , dy 

— = /, ~~ =m. 

dr dr 

I = cos 6, m = sin #, being constant. Therefore 

This is a much simpler way of evaluating this derivative than that 
proposed in § 193. 

As before (see Ex. 11, § 192, § 193), 

tan go = ~ = -£~ cos B 4- -£- sin 8 (2) 

dr dx ay v ' 

is the slope to the horizontal plane of a tangent line to the surface, 

in a vertical plane making an angle 6 with xOz. 

Again, suppose, as in § 194, that x and y are related by 

0(jr, y) = o, and we wish the derivative of f with respect to s, the 

length of the curve (p(x, y) = o, measured from a fixed point to x, y. 

Then, putting / = s in (5) § 194, 

df_^d/dx dfdy 

ds dx ds "^ dy ds W 

dx dy 

But -r = cos 6, — = sin #, where is the angle which the tan- 
ds ds 

gent to <p(x, y) = o at at, _y makes with Ox. Hence we have the 
same value of the derivative as in (1), 

£-£-,+ £-* (4) 



294 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVI. 

which is also the slope to the horizontal plane of the tangent line to 
the surface. 

196. The Total Differential of /(.v, r).— By definition, the 
differential of a function is the product of the derivative into the 
differential of the variable. Hence, mutiplying (5), § 194, through 
by dl, we have for the total differential of /at x, y 

d '=% c dx + % d >- M 



dx=dj, J-dy = d ) f 



Observe that 

dx y dy 

are the partial differentials of/! Hence 

d/=S x /+B y /; (2) 

or, the total differential of/ at x, y is equal to the sum of the partial 
differentials there. 

The value of the differential at a fixed point depends on the 
values of dx and dy, which are quite arbitrary. 

The geometrical interpretation of the differential is as follows: In 
Fig. 123, let dx = MQ and dy = MR. Draw PR', Q'M' y Q" S 
parallel to MR. Then 

3*/ = Q'Q" = M S and d y f=R'R" = SM". 
.-. df- M'S + SM" = M'M"; 

or, the differential of the function is represented by the distance 
from a point in the tangent plane to the surface at P from a hori- 
zontal plane through P. 

197. The Total Derivatives with respect to x and^. — If, in the 

total derivative 

df _ df dx^ ¥^ 

~di ~~d^~dt"^~d^~dt 1 

we take / = x, then the total derivative of/ with respect to x is 

dx ~ dx~T~ dy dx' \ ' 

If we take / = y, then 

<J/ = d/dx df 

dy dxdy "^ dy' { } 

Equations (1) and (2) represent the total derivatives of / with 

regard to x andjy respectively. These derivatives are quite distinct 

and different from the partial derivatives, as is shown by the formula?, 

and as is exhibited in their geometrical interpretations as follows: 

The total derivative of z —f(x, y) with respect to x is the limit 
of the difference-quotient 



Art. 197.] 



T( >TAL 1 HFFERENTIATION. 



95 



x and v varying as the coordinates of a point on some curve MH in 
the horizontal plane. 




Fig. 124. 



If, P x is x 19 y x9 z 19 then, in Fig. 124, 

z x - z = /P 1 = J'P; 9 x x - x = N'M> = J'P'. 

dz 
Therefore — = tan a is the total derivative of z with respect 

to x. That is, the total derivative of/" with respect to x is repre- 
sented by the slope to Ox of the projection P'T' of the tangent PT 
to the surface on the vertical plane xOz. The tangent PT being 
in a vertical plane through P which makes with xOz the angle 

dy 
determined by — = tan 6, as determined from cf){x, y) = o. That 



dx 



4y , 



is, ~ is the slope to Ox of the horizontal projection MN oi the 

tangent PT. 

In like manner the total derivative of/" with respect toy is equal 
to tan /3, this being the slope to Oy of the projection of the same 
tangent PT on the perpendicular plane^6te. 

Equations (1) and (2) are immediately determined from the total 
differential 

¥ 



4T= 9 £* 



dy 



dy 



by dividing through first by dx and then by dy. 
In Fig. 124 we have 

df =JT = J'T' = J"T" 9 

and equations (1) and (2) can be verified by the differential quotients 

taken from the figure directly. 



296 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVI. 

198. Differentiation of the Implicit Function f(x, y) — o. — An 
important and valuable corollary to the total differentiation of the 
function z = /"(.v, r) is that which results in giving the derivative of 
y with respect to x in the implicit function f(x, y) = o. 

Since z — o in z =/[x, y) gives f(x, y) = o, and in f(x, y) = o 
are admissible only those values of x andjy which make z constantly 
zero, the derivative of z with respect to any variable must be o. 

Therefore, from (5), §194, or (1), § 196, § 197, 

K 

dy dx 

dx ~~ ~ ~df' 

¥ 

This has been geometrically interpreted in Ex. 13, Chap. XXV. 

In general, the plane z = c, c being any constant, cuts the surface 
z =y(-v, y) in a contour line, or curve in a horizontal plane, at dis- 
tance c from the horizontal plane xOy. The equation of this curve 
in its plane \sf(x, y) = c. In the same way as above, 

o. 



¥ 

dx 


dx 
~dt 


d/dy 
~*~ dy dt ~ 


dt ' 


dc 
~ It 








dy 




¥ 




•'■ 


dy_ 
dx 


_ dt 
~~dx 

~di 


= - 


dx 

' ¥ 

dy 



which corresponds to the slope of the tangent to the contour (at the 
point x y y, c) to the vertical plane xOz. 

EXERCISES. 

1. if jc- 3 + y 3 — z ax y = c i ^ n< ^ Dx}'- 

Here *L = S{x * - ay), ¥= S (y* - ax) 

dy _ x 1 — ay 
dx ~ ax — y 1 ' 



dy y log y* - y 



2. Find D x y in x m /a** -f y**/6** = 1 

3/ _ tnx™-* df _ wj""- 1 
dx ~ a m ' dy ~ h™ 

3. If x log y — y log x — o, then 

ax x log xy — x 

4. Let x = f> cos 6. Find the total differential of x. 

-— = cos Q, — - = — ft sin 9, 
dp OQ ' ' 

.*. dx = cos 6 dft — ft sin dQ. 



Art. 19S.J TOTAL DIFFERENTIATION. 297 

5. Find the slope to the horizontal plane of the curve 

1 =x+y. J 

6. Find the slope to xOy (the steepness) of the curve cut from the hyperbolic 
paraboloid % — x 2 /d i — y 2 /b 2 by the parabolic cylinder y 2 = 4/>x. 

We have 

dz dz dx , dz dy 

tan co = — = - 

ds dx ds ' dy ds' 

s being the length of the parabola y 2 = \px. Here 

dz 2x dz iy 

dx ~ a 2 ' dy ~ b 2 ' 



*-[■+»"] 



tan go 




VP + 

which is the declivity of the curve in space at x, y, z. 

Find the points at which the tangent to this curve is horizontal. 

7. If u = tan-'(//x), du = (x dy - y dx)/(x* -\- y 2 ). 

8. If z = xy, dz = yxy-i dy -f- xy log x dy. 

9. Find the locus of all the tangent lines to a surface z — f(x, y) at a point 
(a, b, c), P. 

Through /'draw a vertical plane, Fig. 123, rMP, whose equation is 

x — a y — b 

■ — 7— = = r. (1) 

I m ' 

Then the equation to the tangent line, PM", to the surface at P, in the plane 
rMP in terms of its slope at a, b, c, is 

z- c _ df 

r ~ dr' 
z and r being the coordinates of any point on the tangent line. But at a, 6, c 
df _ d/O, b) da df{a, b) db 

dr ~ da dr db dr 

Therefore the equation to the tangent line to the surface at a, i>, c, whose hori- 
zontal projection makes / with Ox (where / = cos 6, m = sin 6), is 

Z - C = rl ^ + rm db- ^ 

Eliminating rl and rm between (1) and (2), we have 

"-' = e-4t£ + Cr-»)fr (3) 

an equation of the first degree in x, y, z, which is the locus in space of the tangent 
lines at a, b, c on the surface. This locus is a plane, Exercise 1, Chap. XXIV, 
touching the surface at a, b, c, and is defined to be the tangent plane to the surface 
at a, b, c. 



298 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVI. 

10. Show that the equation to the tangent plane to the surface z = ax 2 + by 2 
at any point x', y', s' on the surface is 

z -\- z' = 2{axx' -\- byy'). 

11. Use the equation to the tangent plane 

to verify Ex. 12, § 192. 

The direction cosines of the plane are proportional to — , — , — I. Hence 

if /, m, n are these cosines, 

/ _ m n 1 

V ~ ¥ = 

da db 



1 V I+ © ,+ ® 



Also, sec 2 ^ = i/« 2 , giving the same result as Ex. 12. 
12. Show that when 

3/ a/ 

d~x = °' fy = °' 



(1) 



the tangent plane to the surface is horizontal at values of x,y satisfying z =f(x,y) 
and (i). 

13. Show that the curve on the surface z =/(x, y) at all points of which the 
tangent plane to the surface makes the angle 45 ° with xOy is the curve cut on the 
surface by the cylinder 



©"+ (I)'- 



14. Apply Ex. 13 to show that the cylinder x 2 -f- y 2 — \a 2 cuts the sphere 
x 2 -\- y 2 -\- z 2 ■= a 2 in a line at every point of which the tangent plane to the sphere 
is sloped 45° to the horizontal plane. Draw a figure and verify geometrically. 

15. The equation x 2 -j- y 2 = a 2 represents a vertical cylinder of revolution whose 
axis is Oz and radius is a. Find the equations of the path of a point which starts 
at x = a, y = o, z = o and ascends the cylinder on a line of constant grade k. 
This curve is the helix, a spiral on the cylinder, having for its equations 

z . z 

x = a cos -j- , y = a sin j— 
ka' ' ka> 



CHAPTER XXVII. 

SUCCESSIVE TOTAL DIFFERENTIATION. 

199. Second Total Derivative and Differential of z = /(x t y). 
It has been shown in § 194, (5), that 

dt dx dt T dy dt ' {I) 

where x and y are any differentiate functions of/. 
If we differentiate again with respect to /, then 



d 2 / _ d /df dx\ d /df dy 
~di 2 ~ df\dx~ ' dt)^"di [dy "dt 

~T7t[dx)^~dx dfi '^~dtdt\ey)^~dydt 2 



(2) 



Since — , ~ are functions of x andj' to which (1) is applicable, 

in the same way we have 

d /df\_ 9 /df\ dx H(¥\ 4y 
Jt [dx) ~ dx \dx) ' ~df ~*~ dy \dx) ' ~di ' 

~ dx 2 dt + dy dx dt ' W 



d L (d/\ _ a 2 / dx , a 2 / dy 



(4) 



Also, 

dt \dy J ~ dx dy dt ' dy 2 dt 
Substituting in (2) and remembering that 
a 2 / _ d*f 

dx dy dy dx* 

we have finally for the second total derivative of f(x,y) with respect 
to/ 

<Pf_&f(dx\ d 2 f dx dy d 2 f / dy\* df d 2 x df <Py 

dt 1 ~ bx 2 \dt) + 2 dx dy Tt dt + ay \dt) + dx df 1 + dy dt 2 ' (5) 

Multiplying through by dt 2 , we have the second total differential 
of f(x,y), 

299 



300 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cn. XXVII- 

In (5), /is taken as the independent variable, and while dt is per- 
fectly arbitrary in (1) in actual value, we agree, as in Book I, that dt 
shall be taken as having a constant value in the successive differen- 
tiations. 

Thus if we take x as the independent variable instead of /, then dx 

■ , • , • i d * x d /**\ 

is taken constant, in which case -— ; =- -7- [-—)= o, and we have for 

ax 1 dx \dx] 

the second total derivative of/* with respect to x 

dy = dV jy * , ?¥/±\\ V^r M 

dx 2 dx*'*' dx dy dx~^ ty \dx) ^ dy dx* K1) 

In like manner takings as the independent variable, changing / to y 
in (5), we have dy constant, and the total derivative of/" with respect 
to y is 

dy_ dy (dx_v jv to**/* V<^ (9 . 

df dx 2 \dy) "^ dx dy dy ~*~ dy*^ dx dy*' K ; 

dty 

200. -~ v when f(x,y) — o. 

The formulae of the preceding article furnish means of expressing 
the second derivative of y with respect to x in an explicit function 
f(x y y) = o, in terms of the partial derivatives off(x,y). This gen- 
erally saves much labor in computing this derivative when/ is a com- 
plicated function. 

For brevity, represent the partial derivatives of/" with respect to x 
andy by 

f f f" f" f" etc 

J x) Jy> /«> J xy> J yy 1 *»*»'• j 

and the first and second derivatives of y with respect to x by y' , y" . 
Putting/" = z = o in (7), § 199, we have 

o =/£ + 2/;;y +/;;y 2 +/;/'. 

Buty = —f'x/fy. Substituting this and solving fory, 

>py _ '/a/i/i -m/jf -/;;(/:r 

In like manner we get, by interchanging x and y, the second 
derivative D'~x. Otherwise deduced from (8), § 199. 

201. Higher Total Derivatives. — We shall not have occasion to 
use the higher total derivatives of z = f{x, y) above the second. 
They, however, are deduced in the same way as has been the second, 
by repeated applications of the formula for forming the first derivative. 
For the third total derivative of/" with respect to / see Exercise 35 at 
the end of this chapter. 



Art. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 301 

The higher total derivatives of/"(*v, y) with respect to an arbitrary 
function / of x and y become very complicated and are seldom 
employed in elementary analysis. There is, however, an important 
particular case in which the higher derivatives of/[x f jr) require to be 
worked out completely — that is, when x and y are connected by a 
linear relation. This case we now consider and call it linear dif- 
ferentiation. 

202. Successive Linear Total Derivatives. — To find the «th 
derivative of/(x,y) with respect tor, when x and y are linearly 
related by 

x — a y — b 

a, b, /, m being constants. 

The first derivative is, as found before, 

£--='!+"£ <■> 

Differentiating again with respect to r, we have 

Otherwise this follows immediately from (5), § 199, wherein 
dx dy d z x d 2 y 

' = '' IF = 1 ' Ir= m < lr>=dF = °- 

Differentiating (2) again with respect to r, and rearranging the 
terms, we have 

2-'S +»* 3i + «•*&+-?■ « 

We observe that (1), (2), (3) are formed according to a definite 
law. The powers of /, m. and their coefficients follow the law of the 
binomial formula. 

, , 9 a ^ J 

If we consider the symbols — , — as operators on/, and write 
conventionally 

dx> dy = \dx) \dy)' 
then we can write 

dy / d d y 



= \ l *Z + m -x- 



dr 2 \ dx dv 

dV (.9 
dr< \ d.r 



)/, . (5) 



302 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVII, 

in which the parentheses are to be expanded by the binomial formula 

d 3 

and the indices of the powers of—— and — — taken to mean the num- 

dx dy 

ber of times these operations are performed. 

We can demonstrate that this law is general and that we shall 
have 

d V _ // a , - d 
dx" 
as follows. 

First, observe that 






(7) 



For 



Also, 



d df _ d df d df _ 3 df 
dr dx dx dr ' dr dy dy dr 

d_df _d 2 f dx df dy 

dr~ dx ~~ dx^ dr + dy dx~dr' 

d df J d / df 3/\ 

dx~~dr~~dx\ dx~ Jr?n ly) i 

.ay . dy 

— * 5—5 + m 



(8) 



dx 2 dx dy ' 

which proves the first equality in (8), and the second is proved in the 
same way. 

Now assume (7) to be true. Differentiating again with respect to r, 
we have 

d H +f 



dr H+ 



f d /d d \« 

- = d-r[ / dx+ m -W) / > 

/ a dydf 



The memoria technica (7) being true for n = 3, it is true for 4, 
and so on generally. 



Art. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 303 

EXERCISES. 

1. Given x 2 -f y* = a\ find Dy. 

2. If -x- 3 + xf - ay* = o. / = (3^ + y*)/2y(a - x). 

3. If (* + „*)' = a^/ 3 - «=). *- **- ~" 2 ) 



4. If z" = 



dt u(a 2 + t 2 -f u 2 )' 

X + Z dz 22 2 



^ — s' </.* 2S.T — «^x 2 — z 2 )' 

3/ — ,— dv 12/a 1 ' 2 4- 2it~ ^ 

5. If « 2 t' 2 + #« - V- = o. y- = — r-^= . 

6. If f{x, y) — o. is the equation to any curve, show that 

are the equations to the tangent and normal at x, y. The running coordinates 
being X, Y. 

7. Show by Ex. 6 that the equations of the tangent and normal to the ellipse 

x 2 /a 2 + y 2 /b 2 = 1 are 

^1+^=1 and a 2 *-b 2 -=a 2 -b 2 . 
a 2 b 2 x y 

8. Show that the second derivative of y with respect to x, in/(x, y) = o, can 
be expressed in the form 

d 2 y _ _ \dx dy dy dx ) J 

iix* ~~ WY ' 

\¥/ 

9. Show that the ordinate of the curve /(x, y) = o is a maximum or a minimum 
when/X = o, according 2isf xx and/ y are like or unlike signed. 

For a maximum value oiy we must have 

dy_ = _df /¥ =0 

dx dx j dy ' 

or f x = o, fy j£ o. When this is the case, by § 200, 

dty = _dy jdf 
dx 2 dx 2 I dy 

which gives a maximum when f xx and/^ are like signed and a minimum when 
unlike signed. 

10. Show that the maximum and minimum ordinates of the conic 

/ = ax 2 -f- by 2 -\- 2/ixy -\- 2gx -\- ?fy -f- d — o 
are found by aid of 

ft = ax -f hy -f g = o. 

If fy = by -{- &* + /", is positive, the ordinate is a maximum ; if negative, a 
minimum. 

11. Find the maximum ordinate in the folium of Descartes. 

j 3 — T,axy -\- x z = o = f(x, y). 



\f' x = — ay + x 2 , \f y = y°- - 



(IX. 



304 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVII. 

Eliminating _j' between/ =0 f' x = o, we have 
x 6 — 2a % x* = o. 

.-. x — o, x = a \/~2. These values give y = o, y = a \T^, For x = o, 
y = o, we have// = o, but for x = a 4/2, y .— a 4/4, we have a maximum j if 
a = -j-, since 

^ 2 



!>' _ d Y / d f _ 2 

f 2 a.r 2 / 9y a ' 



12. If two curves 0(.r, y) = o, ^(.r, j) = o intersect at a point x, y, and if go 
be their angle of intersection, prove that 

tana» = *«#-#»« . 

0* #y + 0y 4>x 

13. Show that two curves <p = o, ^ — o cut at right angles if at their point 
of intersection 

d(pdip d(p 6^ _ 
dx dy dy dx ~~ 

14. Apply this to show that the ellipses 

jC*/ a 2 _j_ fjhil - I, x 2 /a 2 _j_ y2//3 2 = x 

will cut at right angles if a 1 — b 2 = a 2 — /5 2 . 

15. Show that the length of the perpendicular p from the origin on the tangent 
to the curve (p(x, y) = o at x, y is 

_ -r^ -hy <py 

16. Show that the radius of curvature of /(.*■, j) = o at x, y is 



^ = 



[if*) 2 + (/;) 2 :p 



/;;(y;y 2 - v*;/;/; +/;;(/* ) 2 

17. If f(x, y) = o, show that 

dfd*y IWf_ ay dy_\J*y_ (± ,dy_ d\* f _ 

dy dx 3 "^ 6 \dx dy "*" dy 2 dx J dx 2 ~*~ \dx "^ dx dy/ J ~ " 



18. If J 2 = 2xy + a 2 , 


show that 
a 2 d?y 


3<z 2 x 

(J--0 5 ' 


d 2 X 

dy 2 ~ 


a* 


</.*■ _ / — x ' </a: 2 


' (y - x) 3 ' dx* ~ 


/" 



Also, that jc = ± <z are maximum and minimum values of x. 

19. Investigate y = sin (x -\- y) for maximum and minimum^/. 

dy _ cos( -r -f- >') ^ _ — 7 

</x ~~ 1 — cos {x -f- y) ' i/jc 2 — [1 — cos(jf -j- y)~\ 3 

20. If z = jt> 2 — 2xy* -f- 3^, show that 

a s , dz 

*dx+>'dy = SZ ' 

, 3»« , a 2 2 ., a 2 s 

* a a^ + ^a^ + 'V = 2 °*- 



21. If *=<p(y + ax)+il,(y - «*)> 

22. If . = iZJ! . d -l - Z. 



a 2 c _ 2 a 2 c 
aT 2 ~~ a a> T 



x - y ' 3* (.r - ;>) 2 



Art. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 305 

23. If v — nz —fix — mz\ then m -"- -}- n— = 1. 

24. If «=;'*, prove 1^ =>*-i(i + log j*) = u 'j x . 

25. If z = \/x^fJ 2 , Prove ^|- + y ~\ \ = o. 



If 8 = j/x 3 -j-;< 3 , prove Lr_ -f/_j 3 = 



26. The curve x 3 -\~ y 3 — 3.1 = o has a maximum ordinate at the point 
I, \/2, and a minimum ordinate at — 1, — 4/2. 

27. The curve p(sin 3 -|- cos 3 0) = a sin 2O has a maximum radius vector at the 
point a \^2, \it. 

28. The cur 

maximum ordinate at the point — 4-, 2. 

29. The curve x K -\-y i — $xy % — 2 = has neither maximum nor minimum 
ordinate. 

30. Show that (o, 2) gives y a maximum, and ± \ 4/3, — £ a minimum, while 
I 4/3, I makes x a maximum, and — | 4/3, | gives x a minimum in the cardioid 

(x 2 -f y 2 f — 2y(x 2 + j 2 ) — x 2 = o. 

31. In x x -j- 2ax 2 y — ay 3 = o, y is a minimum at j*r — ± a. 

32. In 3a 2 / 2 -f- xy 3 -\- ^ax 3 — o, y is a maximum for x = 3a /2. 

33. Investigate the conic ax' 2 -\- 2/ixy -f- by 2 = I, for maximum and minimum 
coordinates. 

34. If R is the radius of curvature of/(jr, y) = o, and 6 the angle which the tan- 
gent makes with a fixed line, show from ds = R dB and 6 = tan— 1 dy/dx, that 

R = (i+Z 2 ) 1 = (^ 2 + ^»)i 

_y" *f 2 _y dx — dy d 2 x 

The first when x is the independent variable, the second when the independent 
variable is not specified and dx, dy are variables. 

35. The third total derivative oif(x, y) with respect to any variable t is 
dy _ (dx 8^ dy d \ 3 , d/ d 3 x df d 3 y 

dfi ~ \di dx"^"dt dy] J + dx~ ~dF ~*~ ~ty ~dt 3 

Fdy d 2 x dx d 2 / ld 2 x dy d 2 y dx\ dydty dy~\ 

+ 3 [fx 2 ~dT 2 It + dx~dy \dt J dt + di 2 ~dt)^~ dy 2 dt 2 ~dt J 



CHAPTER XXVIII. 

DIFFERENTIATION OF A FUNCTION OF THREE VARIABLES. 



203. We are particularly interested here in the differentiation of 
a function 

w =/(x, y, z) 

of three independent variables, for the reason that when w — o we 
have 

f(x, y, z) = o, 
the implicit function of three variables, which can be represented by 
a surface in space, and also because the treatment of the function of 
three variables assists in the discussion of the implicit function of 
three variables. 

We do not attempt to represent geometrically a function w of 
three independent variables. 

However, corresponding to any triplet x = a, y = b, z = c, there 
is a point in space which represents the three variables x, y, z for 
those particular values. 

When, corresponding to any triplet x, y, z, the function /(.v, r, z) 
has a determinate value or values it is denned as a function of 

A\ V, Z. 

The function /"is a continuous function of x } y, z at x, v, z when 
for all values of x lt j\, z, in the neighborhood of x, y, z we have 
the number f(x x ,y xi z t ) in the neighborhood oi/[x, y, z). 

Differentiation of w =/[x,y, z). — Let x, y, z and x 1 ,y 1 , 

z x be represented by two points P, P^ 
in space. Complete the parallelopiped 
PRQP X with diagonal PP X , by drawing 
parallels to the axes through P and P v 
Then in the figure we have the coordi- 
nates of P, (-Vj, r, z), and of Q, (.\\, 
r, , z). Let PP X = Ar, and let /, m, 
oc n be the direction cosines of the angles 
which PP X makes with the axes Ox, 
FlG - I2 5- Or, Oz, respectively. 

Then we have 



306 



204. 




x l 


— 


X 


= 


Mr, 


y\ 


— 


V 


= 


?nJr, 


z. 


— 


z 


— 


nAr. 



Art. 205.] FUNCTIONS OF THREE VARIABLES. 307 

Applying the theorem of mean value for one variable, letting 
sr, r. x in succession alone vary, we have 

Av?J -Aw) = (*, - «1/K*^i0. 

/(w) -A* j*) = 0\-J')A(- x \vz), 
A x iJ' z ) - A*y z ) = (*i - *1/1(£j«)i 

where £, j', z; x\, ?/, 0; ^, y 1} C, are points such as Z, M, A r , 
respectively, on the segments PR, RQ, QP V By addition, we have 

Now let / be any differentiate function of x, y, z, such that 
/— f x , when x, y, z become x 1} y x , z v Then for the difference- 
quotient of w with respect to /, 

-f-zrr =ft(&*)-4i +f^ x ^-Jt +S&*iS&T/' 

, . . , . . dw dw dw 

If now the partial derivatives — -, — — , — — are continuous func- 

dx dy dz 

tions throughout the neighborhood of x, y, z, we have, on passing to 

limits in the above equation, the total derivative of f with respect 

to /, 

dt dx dt ~^ dy dt^ dz dt' { } 

The process is obviously general for a function of any number of 
variables, and if F is a function of n independent variables i\ , . . . , 
v n , then the derivative of F with respect to /, a function of these 
variables, is 

dF _ V^dF dv r 

~df ~ / , dzy ~df m 
1 
Second Total Derivative of w =/(x, y, z). — We can differ- 
entiate (1) with respect to / and obtain in the same way 

d v _ l dx d d y d dz d \ V d f d2x 1 d f d2 y \ d l_ d2z 

~d¥~ \Yt^ Jr dtdy'^didz)^~^dx~dt^ + tydf>^~dzdT r ^ 

205. Successive Linear Differentiation. — Of chief importance 
are the successive linear total derivatives oif(x, y, z) with respect to 
r when 

x — a y — b z — c _ 

I m n 

where a, b, c, I, m, n are constants. Then 

x = a -j- Ir, y = b -f- mr, z = c -f- «r, 
and 

dx ■ , dy dz 

dr dr dr 

are constants, their higher derivatives are o. 



308 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVIII. 
Equation (i), § 204, becomes 

¥ = l f + •¥• + » d f « 

dr ox ay dz 

We can differentiate (1) again with respect to r and get 
d*f / t d d d \2 

or obtain the result directly from the equation (2) in § 204. 

We can show, as for two variables, that the «th linear total 
derivative can be expressed by 

3M£+ 4 +■£)>. o) 

where the parenthesis is to be expanded by the multinomial theorem 
and the exponents of the operative symbols indicate the number of 
times the operation is to be performed on/". 

EXERCISES. 

2. If xey 4- log z — yz — o, -- = - v ' . 

dy 1 -yz 

3. If u = log(x 3 + y z + 2 3 - 3-ri's), «L 4- «; + «i = 3(* + >' + s) -1 - 

4. If 7t/ — log (tan x + tan/ -f tan. 2), w ' x sin 2x -\- w' y sin 2y -f- w\ sin 22 = 2. 

5. If w = (x 2 -\-y 2 + s 2 ) - *, show that 

3 2 «/ 9' 2 w 3 2 w _ 

a* 2 " + ~dy z "*" ai 5 

6. If w = e*v, dx °'™ d2 = (1 + 3^ + *y*V- 

7. If re; = .r^ 4 + ^ 2 2 3 +jcy 2 2 2 , 7^; y2 = 6^'S 2 -4- 8j0. 

az 

8. Show that — = 00 at the point (3, 4, 2) on the surface 

* 2 + 3 s2 + xy - V' z ~ 3 X - 4= = o. 

d 2 z 

9. Show that — = o at the point (— 2, — 1, o) on the surface 

4^ 2 + 2 2 - 5*3 + 4>'- +y - 2« — 15 = O. 

10. - \ = — at the point (1 2, — 1) of the surface 
dxd J' 343 

x 2 — y 2 -\- 2z 2 -\- 2xy — \xz -\- x — y -\- z — 5 =0. 

11. Show that the second total derivatives of w = f(x,y, z) with respect to x, y, z 
are respectively 

<Pw _ / d_ dy d_ dz d_\ 2 a/ <Py Of <Pz_ 
dx 2 ~ \dx ~*~ dx dy + dxdz)-'~T~ dy dx 2 "*" dz dx 2 ' 

d*w _ /dx a a r/zay a/ </»* a/^z 
# 2 ~~ w £c + ay + ^ d~z) * + a^< 2 + dz dy* ' 
^ 2 7« _ /dx a ^ a a \ 2 a/rf»* j/"^ ■ 



CHAPTER XXIX. 

EXTENSION OF THE LAW OF THE MEAN TO FUNCTIONS OF TWO 
AND THREE VARIABLES. 

206. Functions of Two Variables. — Let z =f(x,y) be a function 
of two independent variables. 

When x = a, y = b, let z become c = /\a, b). Also, let 

x — a y — b 

-r =— = r w 

Then z = f[x 1 y) = /[a -f- Ir, b -J- w) (2) 

is a function of the one variable r, if a, b, I, m are constants. This 
function becomes c = f(a, b) when r = o. If this function of r and 
its first n -f- 1 derivatives with respect to r are continuous for all 
values of r from r = o to r = r, then, by the Law of the Mean for 
functions of one variable, 



(dz\ _ r* (d^z_ \ _ r w+I ( d* +l z \ 

Here - r - 

\drp ;, 



(3) 



means the /th derivative of 2 with respect to r 



taken at r = o, and I n+j J means the (n -\- i)th derivative of 2 

with respect to r taken at some value u of r between o and r. 

Also, since these derivatives are linear derivatives of z, we have 

d>z\ dt>f{x,v) . , 

when x — a, y = 0, 



dr*h> drf 



= ('as. + "»)*'• *>■ 

since / = (jtr — a)/r, m = ( y — b)/r, from (1). Hence 

3°9 



310 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIX. 



In like manner let x = B,. y 



when r = ct; Z, r/ being num- 



bers respectively between a and x, b and_^. Then 

Substituting the values of (4) and (5) in (3), we have the Law of 
the Mean Value extended to functions of two variables, or 

b - n 



+ nrihy.U x - a ^+ 



"-<} 



/(*». (6) 



207. The geometrical interpretation of § 206 is as follows : 
Given the ordinate to a surface at a particular point a, b, and the 

partial derivatives of the ordi- 
nate at that point. To find the 
ordinate at an arbitrary point 
x,y. 

Let z =f(x,y) be the equa- 
tion to a surface on which 
A, (a, b, c) is the point at which 
the coordinates and partial de- 
rivatives of z are known. Let 
P be the point on the surface 
at which x, y are given and z or 
f(x, y) is required. 

Pass a vertical plane through 
A and P, cutting the surface in the curve AP and the horizontal plane 
in the straight line BM, whose equation is 




Fig. 126. r 



x — a 



= r. 



The equation of the curve AP cut out of the surface by this verti- 
cal plane is 

z = f(a -j- Ir, b -f mr) , 

referred to axes Br, Bz' and coordinates r, z, in its plane rBz' The 
law of the mean is applied to this function of the variable r, resulting 
in (3). Then, since these derivatives are linear, they can be ex- 
pressed in terms of the partial derivatives of z at a, b, and (3) is trans- 
formed into (6). 

208. Expansion of Functions of Two Variables. — Whenever 
the function (2), § 206, of the one variable r can be expanded in 



Ari. 2io] EXTENSION OF THE LAW OE THE MEAN. 311 

powers ofr by Maclaurin's series as given in Book I, then we can 
make n = 00 in (6), and we have 

A *' y) = Zk \ (x ~ a) ^ + °' - 6) is } A "' b) ' 

and the function /\x t y) can be computed in terms off{a, b) and the 
partial derivatives at a, b. 

209. Functions of Three Variables. — Following exactly the same 
process as in § 206, for 

we have the law of the mean for three variables, 



where £, r/, C are the coordinates of some point on the straight-line 
segment joining the points in space whose coordinates are x,y, z and 
a, 5, c. 

Whenever the function of one variable r, 

f[a -j- Ir, b -\- mr, c -j- nr), 

can be expanded in an infinite series of powers of r by Maclaurin's 
series, Book I, then we can make n = 00 in (i), and have 

A*,y,')=£fy {(— 4a+^- b ^-4c} ^'^ (2) 

p = o 

210. Implicit Functions. — The law of the mean enables us to 
express the equation of any curve or surface in terms of positive 
powers of the variables, and permits the study of the curve or surface 
as though its equation were a polynomial in the variables. 

Thus if z =f(x,jy) is constant and o, then f(x,y) = o is the 
equation of a curve in the plane xOy. The equation of any such curve 
can, by (6), § 206, be written in the form 



c 



fi = o 

In like manner, by (1), §209, the equation to any surface 
f(x, y, z) = o can be written 

n 

° =Zh \ ( x -4a+^-4i+(°-4c } Aa > b > c)+R - (2) 



312 FRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIX. 

jR n being (5), § 206, for equation (1) above, and the correspond- 
ing value in (1), § 209, for equation (2). 

211. The law of the mean as expressed in this chapter is funda- 
mental in the theory of curves and surfaces. It permits the treatment 
of implicit equations in symmetrical forms, which is a far-reaching 
advantage in dealing with general problems whose complexity would 
otherwise render them almost unintelligible. 

A most useful form of the equations for two and three variables is 
obtained by putting 

x — a = h, y — b = k, z — c = /, 

and in the result changing a, b, c into x, y, z. 
Thus for two variables 

fix + h,y + k) = £^(* £ + *J) A*,**- (1°) 



For three variables 



ST 1 1 / 



A*+h>y+hz+i)^2^ 7]\ h dx+ ^ +/ a^ 



EXERCISES. 

1. Show that the equation of any algebraic curve of degree n can be written as 
either 

or 

n I / d d \ r 

0= r?M**> +,1 >*) /{0 ' 0) ' ^ 

2. Show that any algebraic surface of »th degree can be written in either of the 
equations 

= £ij j(» -.)! + (, -*) 1 + (—')£ \ r A«. *.0. (i) 

r = o v 

a 

)• (2) 



(x 1- y — } fix, y) is called a concomitant of/(x, y). 
dx dy] 



3. The functioi 



Find the concomitants of a homogeneous function /(x, y) of degree n. 
In (10), § 211, put h = gx, k — gy, then 

.A* + **. y + £>') = 2^- (* s + ;' ^j /(-*-, ;')• 

Since/ is homogeneous in jc and r of degree », 

/(* + gx, y + *r) =/ {(1 + *)*. ( J + *)* } = (* + *)Vl*. jO. 

pr / 9 9 \ r 



Art. 211.] EXTENSION OF THE LAW OF THE MEAN. 313 

This equation is true for all values o£g including o. Therefore, equating like 
powers of g, we have 

„ a 2 / 32/ 0-7" 



(*E +'£)">=•" 



In the same way, if/(.r, y, z) is homogeneous of degree n, we find, by putting 
h = gx, k = gp, I = gz in (11), § 211, as above, the concomitants of/(x, ;>', 2), 

/ 9 a a \ r 

(* dJ: + > a? + * lb) 7 = *" -!)•••(«--+ 0/, 

for r = I, 2, . . . , ». 

The concomitant functions are important in the theory of curves and surfaces. 
They are invariant under any transformation of rectangular axes, the origin 
remaining the same. 



CHAPTER XXX. 
.MAXIMUM AND MINIMUM. FUNCTIONS OF SEVERAL VARIABLES. 

212. Maxima and Minima Values of a Function of Two Inde- 
pendent Variables. 

Definition. — The function z = f(x, y) will be a maximum at 
x = a, y = b, when f (a, b) is greater than_/(jt-, y) for all values of x 
and y in the neighborhood of a, b. 

In like manner/^, b) will be a minimum value of/"(.r, y) when 
/{a, b) is less than f(x , j/) for #// values of .v, v in the neighborhood 
of tf, b. 

In symbols, we havey^tf, b) a maximum or a minimum value of the 
function f{x y y) when 

is negative or positive, respectively, for all values of x, y in the 
neighborhood of a, b. 

Geometrically interpreted, the point/*, Fig. 115, on the surface 
representing z =f{x i y) is a maximum point when it is higher than 
all other points on the surface in its neighborhood. Also, P is a 
minimum point on the surface when it is lower than all other points 
in its neighborhood. 

This means that all vertical planes through P cut the surface in 
curves, each of which has a maximum or a minimum ordinate z at P 
accordingly. 

Also, when P is a maximum point, then any contour line LMN, 
Fig. 1 1 5, cut out of the surface by a horizontal plane passing through 
the neighborhood of P, below P, must be a small closed curve; and 
the tangent plane at P is horizontal, having only one point in 
common with the surface in the neighborhood. Similar remarks 
apply when P is a minimum point. 

When the converse of these conditions holds, the point P will be 
a maximum or minimum point accordingly. 

213. Conditions for Maxima and Minima Values of /(x, 1). — 
Let z =/{x, y), x and r being independent. To find the conditions 
that z shall be a maximum or a minimum at x, y. 

I. Any pair of values x\ y' in the neighborhood of x, y can be 
expressed by 

x' = x -\- lr, y' =y ~\- mr, 

3M 



Art. 2I3-] MAXIMA AND MINIMA VALUES. 315 

where / = cos 8, m = sin 8. Then 

z -=-J\x -\- Ir, y -\- ??ir) 

is a function of the one variable r, if 8 is constant. 

If z is a maximum or a minimum, we must have, by Book I, 

dz d 2 z . . . 

— = o, -— ^ negative or positive, 

respectively, for all values of 8. That is, 

— = cos 6 ~~ 4- sin -^— = o. 
dr dx dy 

This must be true for all values of 8. But when 8 = and 
8 = J.7T, we have 

}&A x >y) = ° and jjA*>y) = ° (0 

respectively. Equations (1) are necessary conditions in order that 
#, _y which satisfy them may give z a maximum or a minimum. But 
they are not sufficient, for we must in addition have 

d 2 z „d 2 f , d' 2 f 3f 

— = l 2 /~ + a^^-4- + »'ft. (2) 

dr* ox 2 ax ay ay 1 v ' 

different from o and of the same sign for all values of 8. When (2) 
is negative for all values of 8, then z at x,y is a maximum; and when 
(2) is positive for all values of 8, then z is a minimum. 

Pot %=A, ^L=H, % = S. 

dx 2 ' dxdy dy 2 

The quadratic function in /, m (see Ex. 19, § 25), 

Al 2 + 2Hlm -f Bm 2 , (3) 

will keep its sign unchanged for all values of the variables /, m, pro- 
vided 

AB - H 2 

is positive. Then the function (3) has the same sign as A. 

(a). Therefore the function f(x, y) is a maximum or a minimum 
at x, y when 

df _ df _ d 2 fdy /dvy_. (A 

. , 9 2 / a 2 / . 

and is a maximum or a minimum according as either -^ or -^, is 

negative or positive respectively. 

(b). If AB — Zf 2 = — , then will (2) have opposite signs when 
m = o and m/l = — A/H\ also when / = o and m/l — — i//i?. The 
function cannot then be either a maximum or a minimum (see Ex. 
19, § 25). 



316 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX. 

(c). If AB — ZT 2 = o, and A, B, H are not all o, then the right 
member of (2) becomes 

(/AJ-jnBy _ {rnB + lHf 
A = B 

and has the same sign as A or B for all values of 6, except when 
vi J I = — A/H. Then (2) is o. This case requires further examina- 
tion, involving higher derivatives than the second; as also does the 
case when A, B, Zfare all o. 

To sum up the conditions, we have _/*(.*•, y) a maximum or a mini- 
mum at x, y when 



/i = o, /; = 



Or f'yy = T 



max. 
min., 
max. 

min.. 



J xx Jxy 
Jxy Jyy 



+ • 



If the determinant is negative, there is neither maximum nor 
minimum; if zero, the case is uncertain.* 

To find the maximum and minimum values of z = f(x, y), we 
solve f' x = o, f' y = o, to find the values of x, y at which the maxi- 
mum or minimum values may occur, then substitute x, y in the 
conditions to determine the- character of the function there. 

The value of the function is obtained by either substituting x, y 
inf(x, y), or by eliminating x, y between the three equations 

^ =A x >J)> /'* = °> f'y = ° 

for the maximum or minimum value z. 

This method employed for finding the conditions for a maximum 
or a minimum value of z =f(x, y) has been that which corresponds 
geometrically to cutting the surface at x, y by vertical planes and 
determining whether or not all these sections have a maximum or 
a minimum ordinate at x, y. 

II. Another way of determining these conditions is directly by 
the law of mean value. We have 

A *.S) -A*,y) = (*' - -) a ^ + (/ --r)^- 

For all values of x', y' in the neighborhood of x, y we have £,, rj also 
in the neighborhood of x, y. If the values/"',., f' y are different from 
o, then the values/"^, f'^ are in the neighborhoods of their limits and 
have the same signs as those numbers for all values of x', y' in the 
neighborhood of x, y. Therefore the difference on the left of the 
equation changes sign when x' — .v, as y' passes through;-, iif y 9* o. 
In like manner this difference changes sign whenj'' =y, as x' passes 

* For examples of the uncertain cast.- in which the function may be a maximum, 
a minimum, or neither, see Exercises 22, 25. at the end of this chapter. 



Art. 213.] MAXIMA AND MINIMA VALUES. 317 

through x, \if' x ^ o. Hence it is impossible f or fix, y) to be a 
maximum or a minimum unless f' x = o and f y = o. 
When f' x — o, fy — o, we have 

A v'.y)- A .v ) ,.) = (.v'-.v)^ +2 (.v-.v)C,'-,) fl ^ /+(l .-,,^. 

If the member on the right of this equation retains its sign unchanged 
for all values of x' , y' in the neighborhood of x, y, the function will 
be a maximum or a minimum at x, y. But in this neighborhood the 
sign of the member on the right is the same as that of its limit, 

(.v - *)»g + >(-' - *)(? -y)^ + (/ -y)%. 

This gives the same conditions as in I, and leads to the same results. 



EXAMPLES. 

1. Find the maximum value of z ~ 2> ax }' — x * ~ }' z - 

This is a surface which cuts the horizontal plane in the folium of Descartes. 
Here 

dz dz 

te = 3<*y-3x 2 , ~= 3 ax- 3 y 2 , (I) 

d 2 z d 2 z c d 2 z 

The equations (1) furnish 

lay - 3 x 2 - o, 3 ax - 3 y 2 = o. (3) 

for finding the values of x, y at which a maximum or a minimum may occur. 
Solving (3), we have 

x = o, y = o, and x = a, y = a. 
For x = o, y = o, 

d 2 z b 2 z I d 2 z \ 2_ 2 

dx 2 dy~ 2 ~ \dx~dy) ~ ~~ 9a ' 

and there can be neither maximum nor minimum at o, o. 
Fur x = a, y = a, 

d 2 2 d 2 z 1 Mr. 



dx* dy 2 \ dx dy) ~ + 2 7 a ' 



d 2 z 
and since — == — 6a, we have the conditions for a maximum value of z at a, a 

dx 2 
fulfilled. Hence at £, a the function has a maximum value a 3 . 

2. Show that a z /2y is a maximum value of 

(« - x )[ a -y)(x+y -*)• 

3. Find the maximum value of x 2 -f- .27 -\-y 2 — ax — by. 

Ans. \{ab — a 2 — b 2 ). 

4. Show that sin x -f- sinj -f- cos (x -*- y) is a minimum when x =y = \n, a 
maximum when x =y = ±7t. 

5. Show that the maximum value of 

(ax + by + c) 2 /(x 2 + y 2 + 1) is « 2 + b 2 + r 2 . 



318 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX. 

6. Find the greatest rectangular parallelopiped that can be inscribed in the 
ellipsoid. That is, find the maximum value of Sxyz subject to the condition 

x 2/ a * +y*/P + z 2 /c 2 = i. (I) 

Let u = xyz. Substituting the value of z in this from (i), we reduce «toa 
function of two variables, 



*"-£-? 



a kg 


hbf 


gf c 



an 2 du 2 

From — = o, — — = o, we find the only values which satisfy the con- 
dx ay 

ditions x = a/ 4/3, y = b/ 4/3. These give z = c/ 4/3, and the volume 
required is 8abc/$ 4/3. 

7. Show that the maximum value of x 2 y^z i , when 2x -f- $y -\- \z — a, is (a/qj 9 . 

8. Show that the surface of a rectangular parallelopiped of given volume is 
least when the solid is a cube. 

9. Design a steel cylindrical standpipe of uniform thickness to hold a given 
volume, which shall require the least amount of material in the construction. [Ra- 
dius of base = depth.] 

10. Design a rectangular tank under the same conditions as Ex. 9. [Base 
square, depth = -J side ol base.] 

11. The function z = x 2 -f- xy -\- y 2 — 5^ — 4^+1 has a minimum for x — 2, 

y= 1. 

12. Show that the maximum or minimum value of 

z - ax 2 -f by 2 + 2hxy + 2gx -f zfy -f c (1) 

is 

I a h 
i h b 
gf c 
\\ e have 

-^ =^ + ^ + ^ = o, -— =A* + *y+/=a (2) 

Multiply the first by x, the second by y, subtract their sum from (i), and we 
get 

«=£*+.# + «• (3) 

Eliminating x and j between (2). (3). the result follows. 

The condition shows that when ab — Ii 1 is positive, the above value of z is a 
maximum or minimum according as the sign of a is negative or positive. If 
ab — k 1 = — , then z is neither maximum nor minimum. We recognize the 
surface as a paraboloid, elliptic for ab — Ji 1 positive, and hyperbolic when 
ab ~ h 2 = -. 

1 3. Investigate z = x 2 -4- 2>. v ~ — x )' ~\~ 3- v — 7. 1 ' ~\~ 1 f° r maximum and min- 
imum values of z. 

14. Investigate max. and min. of x* -\- ) A — x 2 -J- .at —J' 2 . 

x = o, j- = o, max. ; x = y = ± 4, min. ; x = — ^ = ± \ 4/3, min. 

15. The function (jr — v)- — ty{x — 8) has neither maximum nor minimum. 

16. The surface x 2 -f- 2r 2 — 4.x- -4- 4.'' + 3 C ~h J 5 = ° has a maximum 
s-ordinate at the point (2, — 1, —3). 

17. The function x 4 -\- ) A — 2x 2 -4- \xy — 2r- has neither maximum nor minimum 
for x — O, y = o; but is minimum at (-(- 4/2, — 4/2 ), (—4/2, -f- 4/2 ). 



Art. 214.J MAXIMA AND MINIMA VALUES. 319 

18. Show that cos x cos a -f- sin x sin a cos (y — /j) is a maximum when 
X — ex, y = /J. 

19. Show that x' 2 — 6xy 2 -\~ cy* at o, o is minimum if c > 9. and is neither 
maximum nor minimum for other values of c. Hint. Complete the square in x. 

20. Show that (I -f- x 2 -\- y 2 )/(i —ax — by) has a maximum and a minimum 
respectively at 

£ _ j _ 1 ± 4/1 + a 2 -j- />* 

a ~ b a 2 + b' 1 " 

21. Show that 3, 2 make x*y 2 (6 — x — y) a maximum. 

22. Show that a, b make (2ax — x 2 )(2by — y 2 ) a maximum. 

23. Show that 3+44/2 is a maximum, — 6 — 4 4/2 a minimum, value of 

y* - 8j-3 + i8r* - Sy + ** - 3 x 2 - 3 x. 

214. Maxima and Minima Values of a Function of Three 
Independent Variables. 

Let u = /(x,y, z), 

x 1 — x = lr = h, y x —y = mr = k, z x — z = ?ir = g. 
As before, if u is a maximum or a minimum at x,y } z, we must have 
u = y r (~v -+- /r, y -f- «r, -f- nr )> 
a maximum or a minimum for all values of /, ?n, ?i, or 
du du du du 

</r 9a: 9j' 92 

x x — x du y x — y du z x — z du 
r dx r dy r dz' 

must be o for all values of/, m, n or of x v y v z x in the neighborhood 
ofx t y, z, or 

du du du 

& - *) al- + O' ~ y) ay + (2 ' - 2) 9T = °' 

Hence the necessary conditions 

du du du 

5- = o, — = O, — = O. (I) 

Now when the relations (1) hold, and for all values of x l ,y 1 , z x in 
the neighborhood of x, y, z, we also have 

~ = p/z + my;; + «7£ + *i*f'j + ^/~ + ^«/;;, 

or, what is the same thing, 

A/1 2 -f ^ 2 + C^ 2 + 2/^ + 26/^ + 2Hhk ( 2 ) 

(wherein .4 = /£, B = f y ' y >, C=fi', F=f» f G=f'J zi H=f£) 

negative (positive) for all values of h, k, g, then will u be a maximum 
(minimum). 

The condition that (2) shall keep its sign unchanged for all values 



2,20 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cn. XXX. 



of h, k, g has been determined in Ex. 20, § 25, where it is shown that 
when 



A H and A 
H B\ 



A H G 
H B F 
G F C 



are both positive (2) has the same sign as A for all values of h, k, g. 
Therefore /{x y y, z) is a maximum or a minimum at x,y, z, 
determined from 

f x =°> fy' = °> fz = °> 

when we have 



ftl _ ^ max. 

J xx \ mm. 



fxxjyx 

f" f" 

J xy J yy 



= + , 



f f f 

J xx J yx J zx 

J xy J yy J zy 

f" f" f> 

J xz J yz J zz 

The conditions for maximum or minimum can be frequently- 
inferred from the geometrical conditions of a geometrical problem, 
without having to resort to the complicated tests involving the second 
derivatives. 



EXAMPLES. 



/ = X 2 + y-> + Z 2 
fx = 2x—y+i=o, f y = 

■ *=—h y=—h 



- xy. 

O, f z ' = 2Z — 2 = O. 



= I. give / = - f. 

Also, /;; = 2, /;; = 2, f z ' z = 2, /j = - 1, /£ = , /;; = o. 



2 1 

1 2 



2 I o 
120 



002 
Therefore — 4/3 is a minimum value of/". 

2. Find the maximum and minimum values of 

ax 2 -f- by' 1 -f- cz 1 -\- 2fyz -f- 2gxz -j- 2/z.rj' -|- 2«-r -f 2WJ/ -(- 2Ws -|- d. 

Here /*' = 2(<;x -f- >#y -|- 5a -f- u) = o, ' 

// = 2(//a- + ^+> + z')=o, 
fz = 2(gx +fy + «r + a/) = O. 

Multiply the first by .*, the second byj, the third by 
subtract the result from the function/. 

... f=ux + vy + wz + d. 

Eliminating x, y, z between (1) and (2), we have 



(1) 
Add together and 

(2) 



/ 



a h g u 

h b f v 

g f c w 

u v w d 



a h g , 
h b f 
gf c 



which is a maximum or a minimum according as 



a — T 



a h I z= -f, 

h b 



a h % 
h b^f 
if c 



= =F , 



the upper and lower signs going together. 



Art. 215.] MAXIMA AND MINIMA VALUES. 



321 



3. Find a point such that the sum of the squares of its distances from three 
given points is a minimum. 

Let x x% y v »!,... x 3 , y. A , z v be the given points. Then 
/= ^ [(* -x r f -(- (y -y r f + (4 - *,.)»], 

/; = 22(X - X r ) =z o = 3 x - 2x r , 

fy = 2 2(y -y r ) = o = zy - 2y r , 
f z = 22{z - 2,) = o = 32 - ^ 2r . 

• •• * = !(*! + *« + * 8 ), ;' = K.i'i + ^ 2 + M ^ = 1^ 4- z 2 + z 3 ). 

The point is therefore the centroid of the three given points. 

fxx —fyy = f'z'z = °\ /^ = /** ~ f'y'z = °- Show that the solution is a min- 
imum. 

Extend the problem to the case of n given points. 

4. If w = ax 2 -j- byx -\- dz 2 -f- Ixy -f- wyz, show that x — y = 2 = o gives 
neither a maximum nor a minimum. 

215. Maximum and Minimum for an Implicit Function of 
Three Variables. — To find the maximum or minimum values of z in 

f(x, y, z) = o. 
Since the total differentials of/" are o, we have 

d y=( dx l+ d ^+ d 4)^+% d ' x +P^ dh =°- < 2 > 

Also, at a maximum or a minimum value of z we must have 
¥ , ¥ , 

k dx + iy d y 

dz = ^ = 

dz 

for all values of dy and dx. It is therefore necessary that 

¥ ¥ ¥ 

dx=°> 8y-=°> dz*°' (3) 

Substituting these values in (2), we have at the values of x,y which 
satisfy (3), and make dz = o, 

ft - , dx ^ dy S) %/ 

dz- ^ , 



In order that this shall retain its sign for all values of dy and dx, 
we must have 

/£/£' - (/^') 2 = +• (4) 



322 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX. 

Then the sign of d 2 z is that offj/x*. (See Ex. 19, p. 31.) 
Hence z will be a maximum (minimum) at x, y, z, determined 
from 

fx = °> fy' = °> /= °> 

when/" 2 y^ is positive (negative), provided (4) is true. 

EXAMPLES. 

1. Find the maximum and minimum of z in 

2x 2 -f- 5/ 2 -f- z 1 — $xy — 2 X — 4}> — I = o. 

fx = 4* - 4;' -2 = 0, /; = 10,' - 4* - 4 = o, 

give x = f, J = 1, 2f = ± 2. 

/» = 22 = ± 4, /i; = 4, /^/;; - (/gy> = 24. 

s is therefore a maximum and a minimum at |, 1. 

2. Show that z in z 3 -f- 3-r 2 — 4*7 -(- jj's = o has neither a maximum nor a 
minimum at x = — -fa, y = — T | F , z = — ^. 

216. Conditional Maximum and Minimum. — Consider the 
determination of the maximum or minimum value of z =. f{x i y) i 
when x andjy are subject to the condition 0(x,y) = o. 

Geometrically illustrated, z — /(-v,y) and (p(x,y) = o are the 
equations of the line of intersection of the surface z =y~and the ver- 
tical cylinder = o. We seek the highest and lowest points of this 
curve. 

Since, at a maximum or minimum value of z, 

* = j£dx +?L& = o, (1) 

also ^dx-]-—dy=o, (2) 

we have, eliminating dy, dx, the equation 

fx <Py —/}</>'* = ° (3) 

to be satisfied by x, y at which a maximum or minimum occurs. 
Equation (3) together with = o determines x and y for which a 
maximum or minimum may occur. 

I fsually the conditions of the problem serve to discriminate be- 
tween a maximum, minimum, or inflexion at the critical values of x, v. 

The test of the second derivative, however, can be applied as 
follows: We have 

&z = f xx dx* + 2/%; dx dy + /;; df + f x d\x + /; dy, ( 4 ) 

which must keep its sign unchanged for all values of x, y satisfying 
= o in the neighborhood of the x,y also satisfying (3). But we 
also have 

0^ dx* + 20.;; dxdy + 0;; df + 0.; d*x + 0; J\v = o. (5) 






Art. 217.] MAXIMA AND MINIMA VALUES. 323 

To eliminate the differentials from (4). (5), multiply (4) by 0, , (5) 
by /J, and subtract, having regard for (3). In the result substitute 
for dy/dx from (2). 

When this is negative (positive) we have a maximum (minimum) 
value of »2. The form of the test (6) is too complicated to be very use- 
ful, and it is usually omitted. 

EXAMPLES. 

1. Find the minimum value of x 2 -\- y 2 when j and y are subject to the condi- 
tion ax -\- by -\- d = o. 

Condition (3) gives bx = ay. Therefore, at 

ad bd 



y 



- <# + &' J ~ a* + b 2 ' 
we have 

X ^ y a 2 + P ' 

which can be shown to be a minimum by (6). Otherwise we see at once from the 
geometrical interpretations that this value of .r 2 -f- y 2 must be a minimum. 

First. \/x 2 -\- y 2 is the distance from the origin, of the point x, y which is on 
the straight line ax -\- by -f- d = o, and this is least when it is the perpendicular 
from the origin to the straight line, which was found above. 

Second, z = x 2 -\- y 2 is the paraboloid of revolution. The vertical plane 
ax -\- by 4- d = o cuts it in a parabola, whose vertex we have found above, and 
which is the lowest point on the curve. 

2. Determine the axes of the conic ax 2 -\- by 2 -\- ihxy — 1. 

Here the origin is in the center, and the semi-axes are the greatest and least 
distances of a point on the ciu've from the origin. We have to find the maximum 
and minimum values of x 2 -\-y 2 , subject to the above condition of x,y being on 
the conic. 

Let u = x 2 4- y 2 and = ax 2 -f- by 2 -f- ihxy — 1 = o. 
Condition (3) gives 

x ax -\- hy 
y ~~ by -\- hx' 
Multiply both sides by x/y and compound the proportion, and we get 
(a — u~ l )x 4- hy — o, 
hx + (b — u-*)y = o. 
Eliminating x and/, there results 

for determining the maximum and minimum values of «. 

217. The whole question of conditional maximum and minimum is 
most satisfactorily treated by the method of undetermined multinliprc 
of Lagrange. 

The process is best illustrated by taking an example sufficiently 
general to include all cases that are likely to occur and at the same 
time to point out the general treatment for any case that can occur. 



324 PRINCIPLES AND THEORY OE DIFFERENTIATION. [Ch. XXX- 

To find the maximum and minimum values of 

u =f(x,j>, z, w) (rj 

when the variables x, y, z, w are subject to the conditions 

<p(x,y, z, w) = o, ( 2 ) 

ip(x, y, z, w) = o. (3) 

Since, at a maximum or minimum value of u, we must have du = o, 
the conditions furnish 

A dx + A d y + A dz + A dw = o, ) 

<t>x dx + <PyJy + <Pz dz + ^ dw = o,V ( 4 ) 

4>x dx + tp' y dy -f ip> x dz -f i/>£ afc, = o. ) 

Multiply the second of these by A, the third by //, A and jjl being 
arbitrary numbers. Add the three equations. 

(A + A0i + tf>$dx + (/; + \<Pi + My)dy + 

(/; + H' z + Mf*)dz + (/J + A0„i + /^)^ = o. (5) 

Since A and /* are perfectly arbitrary, we can assign to them 
values which will make the coefficients of dx and dy vanish; moreover, 
since equations (2) and (3) connect four variables, we can take two of 
them, say z and w, independent, and therefore dz and dw are arbitrary. 
Consequently, in (5), after assigning A and /j as above, we must have 
the coefficients of dz and dw equal to o. Therefore 

A + A 0i + m>K = o, 1 

A + x & + M>i = °i 

/; + A0; + /</•: = o, f ( 6 ) 

A + *0™ + pK = °- j 

The six equations (2), (3), (6) enable us to determine x, y, z, w y 
A, yu, which furnish the maxima and minima values of u. 

The discrimination between a maximum and a minimum by means 
of the higher derivatives is too complicated for our investigation. In 
ordinary problems this discrimination can generally be made through 
the conditions of the problem proposed. 

EXAMPLES. 

1. Find the maximum value of u — x 2 -f- y 2 -f- z 2 when x, y, z are subject to 
the condition 

(p = ax -f by -\- c z -f- d = o. 

Here we have, as in equations (6), 

i)u . , d(p 

& u ■ i d<P ,-.7 

tf + x ^ = ° = v + M ' 

du , dd> 



ART. 217.J MAXIMA AND MINIMA VALUES. 325 

Multiply by 0, b, c and add. Also, transpose and square. Then 

2{ax -f by + cz) + (a 2 4- b 2 + c 2 )X = - id 4- (a 2 4- 4 2 4- c l )X - o, 
4(* 2 +}' 2 + * 8 ) - (« 2 + * 2 + ^)A- 2 = 4" - (« 2 -f * 2 + < 2 W = o. 
/- d 

|/ fl 2 _j_ £2 + f 2 • 

The problem is to find the perpendicular distance from the origin to a plane. 

2. Find when u = .r- -J- y 2 4- s 2 is a maximum or minimum, .r, jr, z being sub- 
ject to the two conditions 

x 2/a 2 4- y 2 /6 2 4- 2 2 /^ 2 = 1, & 4- my 4- »« = o. 

Geometrically interpreted: Find the axes of a central plane section of an ellip- 
soid. 



Equations (6) give 



ix 
2x 4- A — 4- }xl =0. 



22 

2z + \ — -\- un = o. 



Multiply by jf, _y, z and add. We get A = — «. Therefore 

ua 2 l 2 ub 2 m 2 uc 2 n 2 

lx = — — — , my - 



2(« - a 2 ) ' ' 2(« - 3 2 ) ' 2(z< - <: 2 ) 

Hence the required values of u are the roots of the quadratic 



a 2 / 2 b 2 m 2 



+ 



C- it'- 



ll — a 2 u — b 2 11 — c 2 

3. Find the maximum and minimum values of 

u — a 2 x 2 4- b 2 y 2 4- c 2 z 2 , 
x, y, z being subject to the conditions 

x 2 4- y 2 4- z 2 — I, /.r 4- my 4- «z = O. 
The required values are the roots of the quadratic 

/z/(« _ a 2 ) 4- ;« 2 /(w — b 2 ) 4- « 2 /(« — c 2 ) = O. 

4. Find the maximum and minimum values of 

u = x 2 4- y 2 4- z 2 
when .*, y, z are subject to the condition 

ax 2 4- £y 2 4- cz 2 4- 2/V2 4- 2gxz 4- 2/foy s= 1. 
Geometrically interpreted: Find the axes of a central conicoid. 
The conditions (6) give 

x 4- {ax 4- hy 4- £z)A = o, 

; + (^ + ^+/zU = 0, 
^ + (^4-^4- «W = o- 
Multiply by .*, _y, 2 and add. . •. A = — #. 
Eliminating .r, jy, z from the above equations, 

a — it—*, h , g 

h , b - u-x, f 

g , f , c-u-i 

The three real roots of this cubic, see Ex. 17, § 25, furnish the squares of the 
semi-axes of the conicoid. 



326 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX. 

5. Show how to determine the maxima and minima values of x 2 -j- y 2 -\- z 2 
subject to the conditions 

(*« 4- yl -f S 2)2 _ a 1 x 1 _^ p y 2 _j_ C 2 2 3 f 

Ix -j- my -\- nz = o. 



EXERCISES. 

1. Show that the area of a quadrilateral of four given sides is greatest when it 
is inscribable in a circle. 

2. Also, show that the area of a quadrilateral with three given sides and the 
fourth side arbitrary is greatest when the figure is inscribable in a circle. 

3. Given the vertical angle of a triangle and its area, find when its base is 
least. 

4. Divide a number a into three parts x, y, z such that x m y n zP may be a 

maximum. x y z a 

Ans. —————— 



p m+n+p 

5. Find the maximum value xy subject to the condition x 2 /a 2 -\-y 2 /b 2 = i. 
This finds the greatest rectangle that can be inscribed in a given ellipse. 

6. Find a maximum value of xy subject to ax -J- by = c, and interpret the result 
geometrically. 

7. Divide a into three parts x,y, z, such that xy/2 -\-xz/$ -j-yz/4 shall be a 
maximum. Ans. x/21 = y/20 = z/6 = a/47. 

8. Find the maximum value of xyz subject to the condition 

x i/a 2 -\-y 2 /b 2 -f z l /c 2 = 1 
by the method of § 216. 

9. Show that x -j- y -\- z subject to a/x -f b/y -j- c/z = l is a minimum when 

x/ \/a ' = yj \/b~= zj \/c '= \/~a "-f \Zb~-\- \/7. 

10. Find a point such that the sum of the squares of its distances from the 
corners of a tetrahedron shall be least. 

11. If each angle of a triangle is less than 120 , find a point such that the sum 
of its distances from the vertices shall be least. [The sides must subtend 120 at 
the point.] 

12. Determine a point in the plane of a triangle such that the sum of the squares 
of its distances from the sides a, b, c is least. A being the area of the triangle. 

xyz 2A 



a b c a 2 4- b 2 -f c 2 

13. Circular sectors are taken off the corners of a triangle. Show how to leave 
the greatest area with a given perimeter. [The radii of the sectors are equal ] 

14. In a given sphere inscribe a rectangular parallelopiped whose surface is 
greatest; also whose volume is greatest. [Cube.] 

15. Find the shortest distance from the origin to the straight line. 

Z x x -f m x y +n l z=p l , 

l 2 X + m% y _|_ n%Z — p 2 . 

The equations of the planes being in the normal form. 



Art. 217.] 



MAXIMA AM) MINIMA VALUES. 



We have, if u 2 = x 2 + y 2 -\- z 2 , 

2x + /jA + /> = O, 

2J -f- Wj/l -f- w 2 z/ := O, 

2z -(- «,A -f- «.,/* = o. 

Multiply these by x, y, z in order and add. Multiply by l x , m x , n x in order 
and add. Multiply by / 2 , m 2 , n 2 in order and add. Whence the equations 

2«2 + p x \ -f p^ - O, 

2 P\ + ^ + cos 9 /* = °, 

2 Pi + cos ^ + V- = o. 

Since / x 2 -+- w x 2 -f »j 2 = / 2 2 + w 2 2 + « 2 2 = 1, /^ -f m x m 2 -f »j» 2 = cos 0, where 
is the angle between the normals to the planes. Eliminating X and /i, we have 



A 



u 2 sin 2 



I cos I 
COS I 

»1 2 +A 2 - 2^ 2 COS 



which result is easily verified geometrically as being the perpendicular from the 
origin to the straight line. 

16. A given volume of metal, v, is to be made into a rectangular box; the sides 
and bottom are to be of a given thickness a, and there is no top. 

Find the shape of the vessel so that it may have a maximum capacity. 
If x, y, z are the external length, breadth, depth, 



* + 



V 



3« 



= i*. 



17. Find a point such that the sum of the squares of its distances from the faces 
of a tetrahedron shall be least. If Fis the volume of the solid, x, y, z, w the per- 
pendicular distances of the point from the faces whose areas are A, B, C, D, then 

x _ y _ z __™ _ iv 

A ~ B = C ~ ~ 



A 2 + B 2 + C 2 + D 2 



cz)e~ 



is given by 



c \ I a 2 b 2 c 2 \ 

^z-\j 2 U 2+ ^ + rV 



18. Of all the triangular pyramids having a given triangle for base and a given 
altitude above that base, find that one which has the least surface. 

The surface is %(a -f- b -f- c) 4/r 2 -j- k 2 , where a, b, c are the sides of the base, r 
the radius of the circle inscribed in the base, h the given altitude. 

19. Show that the maximum of (ax + by -i- r*\*-«*** - Py 2 ~ 

a b 

a 2 x fj 2 y y 

20. Show that the highest and lowest points on a curve whose equations are 

<p(x, y, z) = o, ip(x, y, z) = o, (1) 

are determined from these equations and 

<p x ' -f Xip x ' = o, <f)y' -f Aip y ' = o. (2) 

21. Show that the maximum and minimum values of r 2 = x 2 -f- y 2 -f* 2 2 , where 
x, y, z are subject to the two conditions 

ax 2 -f- by 2 -\- cz 2 -\- 2fyz -j- 2gxz -f- 2hxy = I, Ix -j- my + nz = o, 

are given by the roots of the quadratic, 

h 



a — r—2. 

h 

g 
I 



f 



328 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX. 

Geometrically, this finds the axes of any central plane section of a conicoid with 
origin at the center. It also solves the problem of finding the principal radii of 
curvature of a surface at any point. 

The following four exercises are given to illustrate the uncertain case of max- 
imum and minimum conditions. 

22. Investigate z = 2x 2 — %xy 2 -f- y* = (y 2 — x)(y 2 — 2x). 

At o, o we have z' x = z' = z' x ' y = z' y ' y — o, z'J x = 4. The conditions 
z'J x z'' — (zJc'v) 2 = ° makes the case uncertain. The function z vanishes along 
each of the parabolse y 2 = x, y 2 = 2x. It is positive for all values x, y in the 
plane z = o, except between the two parabolas, where it is negative. The function 
is neither a maximum nor a minimum at o, o, since it has positive and negative 
values in the neighborhood of that point. In fact z is negative all along y 2 = 3-*/ 2 
except at o, o. 

23. 2 = a 2 y 2 - 2ax 2 y + x* + y*. 
At o, o the case is uncertain. Put/ = mx, then 

z = x 2 [(l -\- m*)x 2 — 2amx -\- a 2 m 2 ]. 

When x or y is o the function is positive. For all values of /// the quadratic 
factor in the brace is positive. 

Hence z is a minimum at o, o. 

24. z — y 2 — xy 2 — 2x 2 y -\- x*. 

As in 23, the condition is uncertain at o. o. Put j = mx. Then 

z — x 2 [x 2 — m(m -f- 2)x -|- m 2 ]. 

The function is positive when x or y is o. For any value of m not arbitrarily 
small z is positive for all arbitrarily small values of x. But since 

m\\m 2 — \?n — 1) 

is negative for all arbitrarily small values of m, the quadratic function of x in the 
brace has two small positive roots for each such value of 1/1. Between each pair of 
these arbitrarily small roots the quadratic factor, and therefore z. is negative. The 
function is neither a maximum nor a minimum at o, o. In fact along the curve 
x 2 = y the function is z = — x'\ 

25. Consider the function z defined by the equation 

(* - a? + (V x2 -\-y- ~ & = "\ 



or z — a — y2ap — p 2 , 

wherein the positive value of the radical is taken and p 2 z= x 2 -\-y 2 . This is the 
lower half of the surface generated by revolving the circle (x — a) 2 -f- y 2 = a 2 about 
the r'-axis. 
Here 

dz x a — p dz y a — p 



9-* pi \/2a — p d )' pi 4/20 



9 



At all points satisfying x 2 -{- y 2 = p 2 = a 2 these derivatives are o. Also at 
such points 

-xx = * 2 / a *> z yy =)' 2 /a\ zj£ = xy/a\ 
z" z" — (z"\ 2 — o 

• • ~xx "yy K-xy) — VJ- 

The function z is o at each point x, y satisfying x 2 4- y 2 = <* 2 , and is positive for 
every other x. y. It is neither a maximum nor a minimum, nor does it change 
sign in the neighborhood of any x, y in x 2 -f- y 2 = a 2 We shall see later that the 
plane z = o is a singular tangent plane to the surface. 



CHAPTER XXXI. 
APPLICATION TO PLANE CURVES. 

I. Ordinary Points. 

218. We have seen that when the equation of a curve is given 
in the explicit form y = f(x), and the ordinate is one-valued, or 
two-valued in such a way that the branches can be separated, the curve 
can be investigated by means of the derivatives ofy with respect to x, 
or through the law of the mean, as given in Book I, for functions of 
one variable. 

In the same way, when the equation of the curve is given in the 
implicit form F(x,y) = o, we can investigate the curve through the 
partial derivatives and the law of the mean for functions of two variables. 
This amounts, geometrically, to considering the surface z = F(x,y), 
whose intersection with the plane z = o is the curve we wish to 
investigate.* 

219. Ordinary Point. — If F(x, y) — o is the equation to a 
curve, then any point x, y at which we do not have both 

dF A dF 

- — =0 and — - == o 

ox dy 

is called a single point on the curve, or a point of ordinary position, or 
simply an ordinary point. 
By the law of the mean, 

F(,x, j) = F\a, I) + (x -")%■+ O - *) 5£ 

If F(x, y) = o, and a, b is an ordinary point on this curve, then 
F(a, b) = o. Hence 

dF . T , dF 

From this we derive for x( = )a, y( = )b, 
dy _ dF I dF 
dx~ dx I dy' 

*For convenience of notation we shall generally write the explicit equation to 
a curve in the form y =f(x), and the implicit equation as F\x, y) = o. 

3 2 9 



37,o PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. 
Therefore the curve F(x, y) = o and the straight line 

have a contact of the first order at a, b, or (i) is the equation of the 
tangent to the curve at a, b. 

We propose to deduce the equation to the tangent at length, in 
order to lead up to the general methods which are to follow. 

Let F(x, y) = o be the equation to a curve, then 

is the equation to the curve in the form of the law of the mean. The 
straight line 

X-x Y-y 

intersects this curve in points whose distances from x, y are the roots 
of the equation in r, 

o = F {x ,y) +r (l± c+m ^)F+ J (/=J + m ^) V. ( 4 ) 

If the point x, y is on the curve, this is one point of intersection, 
and one root of (4) is o, for F\x,y) = o. 
If in addition we have 

l dx-+ m l)y-= > W 

then two roots of (4) are 0, and the line (3) cuts the curve in two 
coincident points at x,y, and is by definition a tangent to the curve 
at x, y. 

Eliminating /, m between the condition of tangency (5) and the 
equation to the straight line (3), we have the equation to the tangent 
atx,y, 

(*—>£+<'->>£-* w 

the current coordinates being X, Y. 

The corresponding equation to the normal at x> y is 

A' - x r-y 

dF dF ' V* 

dx dy 



Art. 219.] APPLICATION TO PLANE CURVES. 331 

EXAMPLES. 

1. Use Ex. 3, § 211. to show that if F\x, y) = c is the equation to a curve, in 
which F(x, y) is homogeneous of degree n, then the length of the perpendicular 
from the origin on the tangent is 

_ nc 

2. If F[x, y) = u n -\- zt»_ l -f- . . . -f- u x -\- u Q = o is the equation of a curve 
of ;/th degree, in which u r is the homogeneous part of degree r, show that the 
equation of the tangent at x, y is 

dF dF 

X dx + Y dy~ + U ' 1 - 1 + 2Un ~ 2 + ■ ' ' + mt 9 = °- 

If X, Y is a fixed point, this is a curve of the {n — i)th degree in x, y which 
intersects F(x, y) = o in n(n — 1) points, real or imaginary. These points of 
intersection are the points of contact of the n(n — 1) tangents which can be drawn 
from any point X, Y to a curve F = o of the «th degree. 

3. If A', Y be a fixed point, the equation of the normal through X, Yto F — o 
at x, y is 

This is of the «th degree in x, y, which intersects F= o in « 2 points, real or 
imaginary, the normals at which to F = o all pass through X, Y. There can 
then, in general, be drawn n 2 normals to a given curve of the nth. degree from any 
given point. 

4. Show that the points on the ellipse x 2 /a 2 -j- y 2 /b 2 =1 at which the 
normals pass through a given point a, (5 are determined by the intersection of 
the hyperbola 

xy(a 2 - d 2 ) = aa 2 y - (3b 2 x 
with the ellipse. 

5. If F(x, y) = o is a conic, show that its equation can always be written 

» = *.,»>+ |t»— )^0'-*)s}'+*{(*-«>5+Cr-*)oj , «lO 

(a). Show that the straight line whose equation is 

x — a y — b 

— 7- = ~ = r* ( 2 ) 

where / = cos 6, m = sin 0, cuts the curve in two points whose distances from a, b 
are the roots of the quadratic 

o = m.a, t>+r(l± + «|) F+ V (/i. + mff* (3) 

(b). Show that 

dF OF 

is the equation of a secant of which a. b is the middle point of the chord. 
(c). Show that the equations 

dF dF 

— - = O, r— = Q, 

dx dy 

solved simultaneously, give the coordinates of the center of the conic. 

(d). Show that 

x — a y — b . .dF , dF 

— = < and / \- m — = o 

/ m dx dy 



33 2 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. 

are the equations of a pair of conjugate diameters of the conic F = o. whose center 
is a, b. 

6. If k 2 < i, show that the tangent to x 2 /a 2 -j- y 2 /6 3 = k 2 cuts off a constant 
area from x l /a 2 -f- y 2 /b 2 = I. 

7. In Ex. 5, show how to determine the axes and their directions in the conic 
F — o. by finding the maximum and minimum values of r in the quadratic (3). as 
a function of 6, the center of the conic being a, b, 

220. The Inflexional Tangent. — Kx an ordinary point x, y on 
the curve F(x, y) = o, the straight line 

X - x Y - v - 

— = ■ =0 (1) 

I VI 

cuts the curve in points whose distances from x, y are the roots of the 
equation in r, 



o = 

If we have 



l 8x- +m -W = °> ^ 

the line (1) cuts the curve in two coincident points at x,y, and is tan- 
gent to the curve there. 

If, in addition to (3), / and m satisfy 

^+ w !) v =°' (4) 

then the line cuts the curve F = o in three coincident points at x, y, 
provided 

' a a \ 3 

ldx- + m Jy) F *°- W 

In this case the line (1) has a contact of the second order with 
F = o at x,y, and this point is an ordinary point of inflexion. This 
means that the value of l/m — tan 6 in (3) must be one of the roots of 
the quadratic in l/m (4). 

Eliminating / and m between (3) and (4), we have a condition that 
x, y may be a point of inflexion, 

'F» Fp - 2F» Fi F; + F>> F* = o. (6) 

To find an ordinary point of inflexion on F — o, solve (6) and 
F — o for x and y. If the values of x, y thus determined do not 
make both F' x and F' vanish, and do satisfy (5), the point is an 
ordinary point of inflexion. 

The solution of equations (6) and F = o is generally difficult. 

In .general, if x, y is an ordinary point satisfying F = o, and 

/dF a ?£j?Y>- 

\dy dx~. dx dy ' '" °' 



Art. 221.] APPLICATION TO PLANE CURVES. 333 

r = 2, 3, ...,» — i, and 

/dF d OF a 



\3)- 3a- cLv dy) * °' 

then when n is odd we have a point of inflexion at which the tangent 
cuts the curve in n coincident points at x, y. When n is even x t 1 is 
called a point of undulation and the curve there does not cross the 
tangent but is concave or convex at the contact. 

The conditions for concavity, convexity, or inflexion at an ordinary 
point on F— o can be determined as in Book I. For, differentiating 
F = o with respect to x as independent variable, 

° ~~ dx + Jydx' 

JL. ^±\* F < d JL d Jy 

dx "f dx dy J ~t~ dy dx* 

At an ordinary point d x F ^ o or d y F ^ o. Hence the curve is 
convex, concave, or inflects at x, y according as 

(± + ±±Yjr ( d Z±_ d ZAY F 

dy _ \dx ^ dx dy J \ dy dx dx dy J 

dx z ~ dF " /dF\s 

~dy \dy) 

is positive, negative, or zero. 

EXAMPLES. 

1. Show that the origin is a point of inflexion on 

a 3 y = bxy -\- ex* -\- dx 4 . 

2. Show that x — b, y = 2b 3 /a 2 is an inflexion on 

x 3 — 3&r 2 -f- a 2 y = o. 

3. Show that the cubical parabola y 2 = (x — a) 2 (x — b) has points of inflexion 
determined by 3.* -f- a = 4A 

Hint. Solve the conditional equation for (x — a)/{y — b). 

4. If y 2 = f(x) be the equation to a curve, prove that the abscissae of its points 
of inflexion satisfy 

2f(x)f"{x)= \f\x)\ 2 . 

II. Singular Points. 
221. If at any point x, y on a curve F(x,y) — o 

dF , dF 

— - — o and — - = o, 

dx dy 

the point x, y is called a singular point. 

dv dF /dF 

Since -f- = — -r— / -=— > the direction of a curve at a singular point 
dx ox I dy 

is indeterminate. 



334 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. 

222. Double Point If at a singular point the second partial 

derivatives of F are not all o, we shall have 

d 2 F d 2 F d 2 F 

o = (A' - *)• w + 2 (-V - X )(V-y) wj;i + { r-jf ^ . 



Divide through by (A" — x) 2 and let X(=)x. Then 
d 2 F d 2 F ( d\> 



*F d 2 F / dy\ d 2 F/dy\2 

x 2 + 2 dx dy \dx ) + Of \dx ) ' 



This quadratic furnishes, in general, two directions to the curve at 
x, y. Such a point is called a double point. The two straight lines 

d 2 F d 2 F d 2 F 

pass through the point x, y and have the same directions there as the 
curve, and are therefore the two tangents to the curve at the double 
point. 

The coordinates of a double point on F(x, y) = o must satisfy the 
equations 

F=o, F_; = o, F;=o. (i) 

The slopes of the tangents there are the roots t x and / 2 of the 
quadratic 

fiF^+2tF£ + F£=o. (2) 

(A). Node. If the roots of the quadratic (2) are real and different, 
then 

F" F" — F" 2 — — ( A 

the curve has two distinct tangents at x, y, and the point is called a 

node. The curve cuts and crosses itself at a node. 

(B). Conjugate. If the roots of the quadratic in / (2) are imaginary, 

or 

F"x F yy — F xy 2 = +, (4) 

the point is a conjugate, or isolated point of the curve. The direction 
of the curve there is wholly indeterminate. There are no other points 
in the neighborhood of a conjugate point that are on the curve. For 
the equation to the curve can be written 

• = {(*-* \^ + ^-y)r y Y F 
+ i{(*-*> 8 T + ('-■»£}* 

For arbitrarily small values of X — x and Y — y the sign of the 
second member is that of the first term, and (4) is the condition that 



Art. 222.] 



APPLICATION TO PLANE CURVES. 



this term shall keep its sign unchanged. Therefore the equation can- 
not be satisfied for X, Y in the neighborhood of x, r. 
(C). Cusp-Conjugate. If the roots of (2) are equal, or 



7" F" — F" 2 — 

XX ■*■ yy ■*■ X y 



the point maybe either a conjugate point or a cusp. 
one determinate direction there and a double tangent, 
assumes that F xy F ' xx , F y ' y ' are not independently o. 
sideration of the cusp -conjugate class is postponed. 



(5) 

The curve has 
Equation (5) 
Further con- 



Illustrations. 

1. The following example, taken from Lacroix, serves to illustrate the distinc 
tion and connection between the different kinds of double points. 

(a). Let y 2 = (x — a)(x — b){x — c), (1) 

where a. b, c are positive numbers, and a < b < c. 

The curve is real, finite, two-valued, and sym- 
metrical with respect to Ox for a < x < b. It does 
not exist for x < a or b < x < c\ it is finite and sym- 
metrical with respect to Ox for all finite values of 
x > c. The ordinate is 00 when x = co . The curve 
consists of a closed loop from a to b, and an infinite 
branch from c on. The curve is shown in Fig. 127. 
V (b). Let c converge to b. 

Then the loop and open branch tend to come together, 
and in the limit unite in 

/' = (*- a)(x - b)\ (2) 

giving at b a node. (See Fig. 128.) 

(c). Let b converge to a in ( I). The closed oval 
continually diminishes, shrinking to the point a. 








*e^ 



Fig. 128. 
In the limit we have 

y 



j/2 = ( X - af(x - c), (3) 

which consists of a single isolated or conjugate point 
x = a, and an open branch for x > c. (Fig. 129.) 

(d). Let c and b both converge to a. The oval 
shrinks to a, and the open branch elongates to a also, 
resulting in 

y" 1 = (x - af % (4) 

which has a cusp at a. (Fig. 130.) 








Fig. 129. 




2. A clear idea of the meaning of singular points 
on a curve is obtained when we consider the surface 
z = I\x, y), which for any constant value of z is a 
curve cut out of the surface by a horizontal plane. 

For example, using (1), Ex. 1, we have the surface 

z = (x - a)(x - b)(x -c)- y\ 
is symmetrical with respect to the xOz plane, 



Fig. 130. which 

and cuts the xOz plane in the cubic parabola 

z = (x — a)(x — b)(x — c), 
and the horizontal plane in the curve 

j2 - (x - a)(x - b)(x - c). 



336 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. 

A moving horizontal plane cuts the surface in curves of the same family. For 
example, DD is an open branched curve; BB is a curve with a node as in Fig. 128; 
A A is a curve with a closed oval and one open branch as in Fig. 127; so also is CC. 

As the horizontal cutting plane rises until it reaches a maximum point Z"on the 




Fig. ni. 



surface the closed oval shrinks until it becomes the point of contact of the horizon- 
tal tangent plane, which plane cuts the surface again in the open branch T. The 
point of touch T is a conjugate of the curve TT and part of the intersection of the 
surface by the plane. If the cutting plane be raised higher, to a position J, the 
oval and conjugate point disappear altogether and the section is only the open 
branch /. 

Observe that the tangent plane at the node of BB is also horizontal, but the 
ordinate to the surface is there neither a maximum nor a minimum. 

The node of BB is a saddle point on the surface. 

To illustrate the cusp, consider the surface 

z — (x — a) 3 — y 2 . 

This cuts xOz in z = (x — a) z , and the 
horizontal plane in y 2 = (x — a) 3 . All 
planes parallel to yOz cut the surface in 
ordinary parabolse. All sections of the 
surface by horizontal planes are open 
branched curves, none having cusps except 
that one m xOy. All horizontal sections for 
_/£ z negative have inflexions in the plane x = a, 
and their tangents there are parallel to Ox. 
The horizontal sections above xOy have no 
inflexions. As the plane of the horizontal 
section below xOy rises, the inflexional tan- 
gents unite in the tmique double tangent at 
the cusp in the plane xOy. 

3. The above considerations will always enable us to discriminate between a 
conjugate point and a cusp of the first species,* when the singular point is of the 
cusp-con jugate class under condition (5). For, let F[x,y) = o have a point of 
this class, and let F(x, y) = o be the equation of the curve referred to the singular 
point as origin and the tangent there as x-axis. The point is a cusp of the first 
species if F\\\ o) changes sign as x passes through o. If F(x, o) does not change 
sign as x passes through o, the point is either a conjugate or a cusp of the second 
spi cies. If in the neighborhood of such a point no real values of .r, y satisfy the 
equation, the conjugate point is identified. Also, the conjugate points on F = o 
are the values of x, y which make z — F -x maximum or a minimum. 

* A cusp is of the first species when the branches of the curve lie on opposite 
sides of the tangent there. If both branches lie on the same side of the tangent, 
the cusp is of the second species. 




Art. 222,.] APPLICATION TO PLANE CURVES. 337 

The only forms that double points on an algebraic curve can have, 
besides the conjugate point, are nodes and cusps. (See Fig. 133.) 






Node. Cusp, first species. Cusp, second species. 

Fig. 133. 

In fact, all other singular points of algebraic curves are but combi- 
nations of these, together with inflexions. 



EXAMPLES. 

1. Show that the origin is a node of y 2 (a 2 -j- x 2 ) = x 2 (a' 2 — x 2 ), and that the 
tangents bisect the angles between the axes. 

2. Show that the origin is a cusp in ay 2 = x 3 . 

3. Find the singular point on y 3 = x 2 (x -j- a). [Cusp.] 

4. Investigate b(x 2 -\- y 2 ) = x 3 at the origin. 

5. Investigate x 3 — Z ax y -)- JK 3 = o at the origin. 

6. Find the double point of (bx — cy) 2 ■=. (x — af, and draw the curve there. 

[x = a, y = ab/c. Cusp.] 

7. The curve ( y — c) 2 = (x — a) k {x — b) has a cusp at a, c, if a ^ b\ conju- 
gate if a < b. 

8. Investigate y 2 = x(x -j- a) 2 and x& -\- y%~ = a* for singular points. 

9. Investigate at the origin the curve 

F= ay 2 — 2xy 2 -f lyx 2 — ax 3 -j- by 3 -f x* -f ;/» = o. 

Here F x = O, F y = o, F'J 2 — F' x ' x F y ' y — o, at the origin, and the third 

partial derivatives are not all o. The origin is a point of the cusp-conjugate class, 
and j 2 = o is the double tangent. 

Since F K x, o) = — ax 3 -f- x 4 - changes sign as x passes through o, the origin is 
a cusp of the first kind. 

223. Triple Point. — If x, y satisfy the equations 

F=Fi = f; = f>> = f;; = f;; = o, (i) 

and do not make all the third partial derivatives of F vanish. Then 
we have at any point X, Y on the curve 

Divide by {X — x) 3 and make X( = )x. We have the cubic in / 
for finding the three directions of the curve at x, y, 

o = F- x + z tF> x 'J y + Z PF% + fij%. (2) 

The solution of this gives, in general, three values of / = dy/dx, 
furnishing the three directions in which the curve passes through x, y, 



33 8 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. 

which is a triple point on the curve. The equation of the three tan 
gents at x, y is 

9 3)3 



dx ' ' dy 

Some forms of triple points are shown in Fig. 134. 




Fig. 134. 

EXAMPLES. 

1. Show that x* = (x 2 — y 2 )y has a triple point at the origin. 

2. Investigate at O the curve x 4 — $axy l -f- 2a }' i = °- 

224. Higher Singularities. — In general, if 

d r F 

dx» dy* ~ 

for all values of p -f- q = r, and r = o, 1, 2, . . . , » — 1, then 
the curve ^ = o has an //-pie point at x,y, and in general passes 
through the point n times. 

The equation of the n tangents there is 

( 9 3 ) n 

Their slopes are the roots of 

{r x + 'sy) F =°- 

Examples of multiple points are shown in Fig. 135. 




Fig. 135. 

EXAMPLES. 

1. Investigate x*> -\- y b = $eufiy % , at o, o, 

2. Investigate (y — x 2 ) 2 = x b . at o, o. 

3. In x 5 -f- bx* — a*y 2 = o, the origin is a double cusp. 

4. Determine the tangents at the origin to 

y 2 = x 2 (i - x-). [ x ± y - o.] 



Art. 225. J APPLICATION TO PLANE CURVES. 339 

5. Show that x* — $axy -\- y* = o touches the axes at the origin. 

6. Investigate x* — ox-v -\- by* = o at o, o. 

7. Show thato, o is a conjugate point on 

ay 2 — .r ! -f- bx 2 = o 
if a and b are like signed, and a node when not. 

8. Show that the origin is a conjugate point on 

y\x 2 — a-) = x*, and a cusp on (y — x 2 ) 2 = x 3 . 

9. Investigate ( r — x 2 ) 2 = x H at o, o, for ;/ < 4. 

10. Investigate {x/a)* -\- {)'//>)* = 1, where it cuts the axes. 

11. Find the double points on 

x* — 4ax :i -f- 4<2 2 x 2 — b 2 y 2 -f- 2/>'[v — a 4 — 5* = o. 

12. Also on x* — 2ax' 2 y — ax/ 1 -j- a 2 y 2 = o. 

13. Find and classify the singular points on 

x* — 2ax 2 y — axy' 1 -j- a 2 y 2 = o 
when a = I, a >' I, «<i. 

14. Show that no curve of the second or third degree in x and^ can have a cusp 
of the second species. 

Show that if I\x, y) = o is any equation of the third degree, having a point of 
the cusp-conjugate class at the origin and the .r-axis as tangent, the origin is a 
cusp or conjugate point according as F[x, o) does or does not change sign as x 
passes through o, that is, according as the lowest power of x is odd or even. 

15. If F(x t y) = o is any curve of the fourth degree, having at the origin a 
double point of the cusp-conjugate class, and the tangent there as jr-axis, then the 
origin is a cusp of the first species if the lowest power of x in F(x, o) is odd ; 
otherwise, it is a cusp of the second kind or a conjugate point according as the 
co-factor of x 2 in F{x, nix) has real or imaginary roots for arbitrarily small values 
of m. 

16. Show that the origin is a cusp of the second kind in 

x i -\- y* — ay' 6 — 2ax 2 y -j- axy 2 -f- a 2 y 2 = O; 
is a conjugate point in 

x 4 + ) A — a )' % ~ ax 2 y -4- axy 2 -\- a 2 y 2 = O ; 
and a cusp of the first kind in 

•* 3 + j' 4 — a ) ,z — b* 2 )' -f- cx y 2 + a2 y 2 = °- 
225. Homogeneous Coordinates. — The study of homogeneous 
functions is very much simpler thau that of heterogeneous functions, 
owing to the symmetry of the results. This is exemplified in the con- 
comitants. It is therefore of great advantage, in the study of curves, 
to make the equations homogeneous by the introduction of a third 
variable. While we do not propose to follow up this method, it is so 
necessary and so universally employed in the higher analysis that it is 
mentioned here in order to give a geometrical interpretation to the 
meaning of the process and to illustrate what has been said about the 
study of surfaces facilitating the study of curves. 

In the present chapter we have been really studying a curve as the 
section of a surface by the plane 2=0. If now we make the e<]iia- 



34Q PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. 

tion to any curve F(x t y) = o homogeneous in x,y, z, by writing the 
equation 

and clearing of fractions, then we have the homogeneous equation in 
three variables x, y, z, 

F&iy, *) = o. (2) 

/*= o becomes F = o when we make &.= 1. 

But F =■ o being homogeneous in a*, j>, 2, it is the equation of a 
C0«£ with vertex at the origin, and which cuts the horizontal plane 
z = 1 in the curve F = o, which curve is the subject of investigation. 

Consequently any investigation of F x = o carried on for a homo- 
geneous function in a - , j/, 2 is applicable to the curve F — o when in 
the results of that investigation we make z = 1. 

III. Curve Tracing. 

226. In the tracing of algebraic curves, the following remarks are 
important. 

(I). If the origin be taken on a curve of the nth degree, at an or- 
dinary point, the straight line^ = mxca.11 meet the curve in only n — 1 
other points. 

If a curve has a singular point of multiplicity m, and this be taken 
as origin, the Ymey = mx can meet the curve in only n — m other 
points. 

Therefore, if any curve of the »th degree has at the origin a sin- 
gularity of multiplicity n — 2, the line y = mx can meet it in only 
two other points besides the origin, and by assigning different values 
to m we can plot the curve by points conveniently. 

(II). If any curve has a rectilinear asymptote, and we take the 
jy-axis parallel to this asymptote, we lower the degree of the equation 
in y by 1. If there be m parallel asymptotes, and we take the y-axis 
parallel to them, we lower the degree of the equation in y by m. If 
the degree of the equation in y can thus be made quadratic or linear 
in ;', then by assigning different values to x, the curve can be plotted 
by points conveniently. 

(III). In any algebraic equation of a curve F= o, when the 
origin is on the curve, the coefficients of the terms in x,y are the re- 
spective partial derivatives of the function Fat o, o. Therefore the 
homogeneous part of the equation of lowest degree equated to o is the 
equation of the tangents at the origin. The origin is a singular point 
whose multiplicity is that of the degree of the lowest terms ; it is an 
ordinary point if this be I. 

(IV). The Analytical Polygon. — Newton designed the follow- 
ing method of separating the branches of an algebraic curve at a 
singular point, and tracing the curve in the neighborhood of that 



Art. 220.J 



APPLICATION TO PLANK CURVES. 



34i 



point. The method also determines the manner in which the curve 
passes off to 00 . 

Let F(x t y) beany polynomial in x andjr which contains no con- 
stant term. Then 

F(x t y) = 2 C^y = o 

is the equation of a curve passing through the origin. 

Corresponding to each term 
Cr-x-y 1 , plot a point with reference 
to axes Op, Oq, having as abscissa 
and ordinate the exponents p and q 
of .v and y respectively. Thus lo- 
A lf . . . , A 10 , draw 




Fig. 136. 



the simple polygon A l A z A h A^A % A l(i A l 
in such a manner that no point 
shall lie outside the polygon. 

Such a polygon is determined by 
sticking pins in the points and 
stretching a string around the sys- 
tem of pins so as to include them all. 

The properties of the polygon are :* 

(1). Any part of the equation F = o, corresponding to terms 
which are on a side of the polygon cutting the positive parts of the 
axes Op, Oq, and such that no point of the polygon lies between that 
side and the origin, when equated to o is a curve passing through the 
origin in the same way as does F — o. 

Thus, if we strike out of F = o all terms except those correspond- 
ing to terms on the side A Z A 5 , we have left a simple curve which 
passes through the origin in the same way as i^= o. In like manner, 
if we strike out all terms save those corresponding to points on the side 
A X A % , we have another simple curve passing through the origin in the 
same way as does F = o, and so on. 

(2). Any part of the equation F=o corresponding to points 
which lie on a side of the polygon cutting the positive parts of the 
axes Op, Oq, and such that no point of the polygon lies on the opposite 
side of this line from the origin, when equated to o gives a simple curve 
which passes off to infinity in the same way as does F= o. 

Thus the part of F = o corresponding to the side A^A % gives 
such a curve. Again, the part corresponding to A 8 A 10 gives another 
such curve. 

(3). Any side of the polygon which cuts the positive part of one 
axis and the negative part of the other merely gives one of the axes 
Ox or Oy as the direction of an asymptote to F = o, and these are 
more simply determined by equating to o the coefficients of the 
highest powers of x and of jy in F= o. Such a side is A^A^ 

(4). Any side of the polygon which is coincident with one of the 



*For a demonstration of these properties see Appendix, Note 12. 



34 2 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. 



axes Oq, Op, as A X A^> merely gives the points of intersection of 
F = o with Ox or Or accordingly. 

(5). Any side of the polygon which is parallel to one of the axes 
Op, Oq gives rectilinear asymptotes parallel to an axis, or the axis as 
a tangent to the curve according as the side falls under conditions (2) 
or (1). 

EXAMPLES. 

1. Trace jr 6 -\- 2a 2 x s y — Py 3 = o. 
Numbering the terms in the order in which 
they occur, we have A x , A 2 , A 3 , in the polygon 
corresponding to the terms of the equation. 

The curve passes through O in the same 
way as does the curve 
x 6 -\-2a 2 x 3 y = x 3 {x z +2a 2 y)=:0. 
corresponding to A X A 2 , or 
as shown in Fig. 138. 

Also, the curve passes 
through O in the same way 
as does 

2a 2 x'y — b % y* = y(2a 2 x 3 — by 2 ) =r o, 
corresponding to A 2 A Z , as shown in Fig. 139. 

The curve passes off to 00 in the same way as 
does the curve 

6 — ^3 V 3 — n . or x 2 = by, 
Fig. 140. 




Fig. 138. 




x° — oy A = o 
corresponding to A X A % 



The form of the curve is therefore as in Fig. 141, 



Fig. 140. 



Trace the following curves: 
2. x 4 — 2ax 2 y — axy 1 -\- d l y 

4. ay 2 — xy 2 — 2yx 2 -\- ax 4 — x 2 = O 

5. x* - o 2 xy _|_ b 2 y 2 — o. 
7. x* — laxy 2 -f- 2ay z - O. 
9. x* -f a 2 xy — y 4 — o. 

11. a 2 (x 2 +y 2 )-2a(x-y) 3 +x 4 +y 4 =0. 
13. a(y- x) 2 (y + x ) =y 4 + x 4 . 




15. ox(y • 
17. Trace 



^2 _ 



^. 



— ax 3 y — axy 3 -\- a 2 y 



axy -f axy 2 + a 2 y 2 : 
a < I. 



8. x h -2a 3 x 2 -^^a 3 xy — 2a 3 y 2 -\-y 



10. a\x 2 - y 2 ) + x*+y* = 0. 
12. a(y 2 — x 2 ){y - 2x) =y*. 
14. x* — axy 2 -\- y 4 = °- 
16. x* - a 2 xy + b 2 y 2 = o. 
— o, near the origin. 



18. x* — a 2 xy 2 = af. 

20. x* -f ax 2 y = ay 3 . 

22. .r :> +;- 5 = $ax 9 y. 

24. (x - 2\v 2 = 4_r. 

26. {y — x)(y-4x)(y + 2x): 



19. 



,rxy 



Sax 2 



av°. 

21. x(y -xY = b 2 y. 
23. (x - 3lr 3 = (y - i)x\ 
25. (x - i)( x - 2)y 2 = x 2 . 
27. (j, -*)*(;, + *)(, + 2*) 



1 6a 4 . 



Art. 228.] APPLICATION TO PLANE CURVES. 343 



IV. Envelopes. 

227. Differentiation of functions of several variables affords a 
method of treating the envelopes of curves, which in general simplifies 
that problem considerably and gives a new geometrical interpretation 
of the envelope. 

For example, we can supply the missing proof, in § 104, that the 
envelope is tangent to each member of the curve family. When x, y 
moves along a curve of the family 

F(x,y,a) = o, (1) 

a is constant, and we have on differentiation 

But if x, y moves along the envelope, a is variable, and on dif- 
ferentiation of (1) 

dF f dF , dF , 

— dx -f- — - dy -f —~da— o. (?) 

dx ' dy ' da vo/ 

Also, on the envelope - — = o. Therefore -3-, from (2) and (3), 

are the same at a point x, y common to the curve and its envelope. 

228. Again, let a, /3, y be variable parameters in the equation 

F(x,y, a, ft, y) = o, (1) 

where a, f3, y are connected by the two relations 

cp(a, /?, y) = o, (2) <p(a, /?, y) = o. (3) 

We require the envelope of the family of curves (1) when a, /?, y 
vary. Obviously, if we could solve equations (2), (3) with respect to 
two of the parameters and substitute in (1), or, what is the same thing, 
eliminate two of the parameters between equations (1), (2), (3), we 
could reduce the equation to the family of a single parameter and pro- 
ceed as in Book I. This is not in general possible, and it is generally 
simpler to follow the process below. 

Differentiating (1), (2), (3), the parameters being the variables, 



^—da 
da 


+ - 

^ d/3 


*+£ 


dy = 


o, 


60 

^-da 

da 


90 

^ d/3 


*+? 


dy = 


o> 


dip 

^-da 

da 


dip 

"^ d/3 


*+? 


dy = 


0. 



Multiply the second of these by A, the third by //, and add. Deter- 
mine X and pt so that the coefficients of da and d/3 are zero. Then 



344 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. 

if we take dot as the independent variable parameter, the differentials 
d/3 f dy are arbitrary and we can assign them so that the remainder of 
the equation shall be zero. Then 

BF 30 dif; 

dF 30 30 

^ + A a/i + / ^ = ' (5) 

dy+ X d r + M d r = ' < 6 > 

The envelope is the result obtained by eliminating a, f3, y, X, /j. 
between the six numbered equations. 

If we have only two parameters and one equation of condition, the 
particular treatment is obvious; as is also the treatment of the gen- 
eral case when we have n variable parameters connected by n — i 
equations of condition. 

229. We can get a concrete geometrical intuition of the relation 
of curves of a family and their envelope, by letting z be a variable 
parameter and considering 

F(x,y, z) = o 

as the equation of a surface in space. Then the curves of the family 
are the projections on the horizontal plane xOy of horizontal plane 
sections of the surface, obtained by varying z = a. 

EXAMPLES. 

1. Find the envelope of a line of given length, /, whose ends move on two fixed 
rectangular axes. 

We have to find the envelope of 

x/a -f- )'/b = 1 when a 1 -\- b' 2 = I*. 
.-. x/a* = Xa, y/b* = Xb. 
Hence A = a-*, and a = (Z 2 *)*, b = {l*y% 

and the envelope is x* + y 3 = / 3 . 

2. Find the envelope of concentric and coaxial ellipses of constant area. 
Here x'-Jd 1 + y 2 /b°- = 1 and ab = c. 

.-. x */a* = Ab, y*/b* = Xa. .-. 2cX = 1. 
The required envelope is the equilateral hyperbola 2xy — c. 

3. Find the envelope of the normals to the ellipse. 

Here a*x/a - b*y/'/S = a* — b* and a* /a* -f fi*/b* = 1. 

.-. a*x/a* = Xa/a*, b*y//3* = - X/S/b*. .-. X = a* - b*. 



I [en< e 
i;ivr the required envelop 



a _ I ax \ * /? _ I by \ * 

a - \a* - b* ) ' /; "" \<z 2 - b*J ' 



((/ . r) :i + (fry = (<I 2 _ p )z 



Art. 229.] APPLICATION TO PLANK CURVES. 345 

4. Show that the envelope of x/a -f y, b = I, when- a and b are connected by 

»< m vi 

am _|_ ^'» = 6 ->«, is x m ~ l -j-_y«+« — r'"+i. 

5. Show that the envelope of x/l-\-y/m = 1, where the variable parameters 
/, w are connected by the linear relation l/a -f- /////; = 1, is the parabola 



©'+»'■" 



6. Show that if a straight line always cuts off a constant area from two fixed 
intersecting straight lines, it envelops an hyperbola. 

7. Show that the envelope of a line which moves in such a manner that the sum 
of the squares of its distances from n fixed points x r , y r is a constant k 2 , is the 
locus 

^Xr — k 2 , ^X r )' r r ^>*r, X 

2x r y r , 'Sy;. — Z- 2 , 2y r , y 

2x r , 22y r , n , 1 

x , y ,1,0 

Let the line be Ix -f my -j- p — o. Then 

k 2 = l 2 2xl + ™^y*r + np 2 + 2m/ 2y r + 2// 2x r + 2/m 2* r ;>v, 

= ^Z 2 -|- bm 2 + r/ 2 4- 2/w/ -f 2«-// -f- 2/^/w. 

Also, P -\- m 2 — I, / and wz being direction cosines of the line. 
Hence we have 

al -f- /;/// -f gp -f A/ -f 4j^ = o, 

/,/ _|_ bm +fp + Am + I//;/ = o, 

g l 4. //« -\- cp -\- o + i^ =0. 

Multiply by /, m, p in order and add. . •. A = — k 2 . 
Eliminating /, m, p, ju between the equations 

(a _ k 2 )l + hm 4- gp 4- ijux = o, 

^/ 4- (^ _ £2) w 4-/^4- 4//J - o, 
gl 4. / w 4. c p + ^ _ o, 

*/ 4- ^^ 4- / 4" ° = °> 

we have the result. 

8. Show that the envelope of a straight line which moves in such a manner that 
the sum of its distances from n points x r , y r is equal to a constant k, is a circle 
whose center is the centroid of the fixed points and whose radius is one «th the 
distance k. 

Let Ix 4- my -\- p = o be the line. Then P -\- m 2 — 1, and 

k = P2x r 4- n£Ey r -\- np, 

— al -\- bm 4- cp. 
Here we have 

a 4- Ax 4- 2/// = o, 

b 4- Ay 4- 2/z-w = o, 

f 4- A 4- o =0. 

. \ A = — c = — n. Multiply these three equations by /, m, p in order and 
add. Hence k -\- 2/1= o. 



346 PRINCIPLES AND THEORY OF DIFFERENTIATION. |Ch. XXXI. 

The equations a — nx = i/, b — ny = km, squared and added, give the 
envelope 



(*-^)'+(-^y=er- 



9. Find the envelope of a right line when the sum of the squares of its distances 
from two fixed points is constant, and also when the product of these distances is 
constant. 

10. A point on a right line moves uniformly along a fixed right line, while the 
moving line revolves with a uniform angular velocity. Show that the envelope is 
a cycloid. 

11. Show that the envelope of the ellipses x 2 /a 2 -\-y 2 /b 2 =. I, when a 2 -f- b' 1 = k\ 
is a square whose side is k 4/2. 

12. Show that the envelope of line xa™ -\- yb»<- = c™+i, when a n -\-b n = d n , is 



+y 



-m 



13. Find the envelope of the family of paraboloe which pass through the origin, 
have their axes parallel to Oy and their vertices on the ellipse x 2 /a 2 -\- y 2 /b 2 = x. 

[A parabola.] 

14. The ends of a straight line of constant length a describe respectively the 
circles (x ± c) 2 -j- y 2 = a 2 . Show that the envelope of the curve described by the 
mid-point of the line, c being a variable parameter, is 

\{x 2 + y 2 — \a 2 )x 2 -f- a 2 y 2 = o. 

15. Find the envelope of a family of circles having as diameters the chords of a 
given circle drawn through a fixed point on its circumference. [A cardioid. ] 

16. In Ex. 14 show that the area of each curve of the family is %ita 2 when 
c > \a. Also, show that the entire area of the envelope is \a\^Tt — Vz\ 



PART VI. 

APPLICATION TO SURFACES. 

CHAPTER XXXII. 
STUDY OF THE FORM OF A SURFACE AT A POINT. 

230. We shall in the present chapter use f{x,y) and F(x, y, z), 
when abbreviated intrj/"and F, to mean a function of two and three vari- 
ables respectively. 

The functions immediately under consideration are 

z =f(x,y) and F(x,y, z) = o. 

The first expresses z explicitly as a function of x andjr, and is to 
be regarded as the solution of the implicit function F— o with respect 
to z. 

It is to be observed generally that since 

results obtained from the investigation of F — o are translated into 
those for z =fby 

d>+«F &+</ dF _ W+*F _ 

(&Fty~thSty 9 ~dz = ~ *' 'dz^+ r ''~°' 

231. In the present article we recall a few fundamental principles 
of solid analytical geometry which will be needed subsequently.* 

I. The Plane. The equation of the first degree in x, y, z can 
always be represented by a plane. 

We repeat the proof of this from geometry as follows : 

Let Ax + By + Cz -f D = o, (1) 

A, B, C, D being any constants. Assign to x and^y any values x x ,y x 
whatever, and compute^, so that ■x\,y 1 , z l satisfy (1). In like man- 
ner assign arbitrarily x 2 ,y 2 , and compute z 2 so that x 2 , y 2 , z 2 also 
satisfy (1). 

Represent, as previously, x,y, z by a point in space with respect to 

* For a more detailed discussion, see any solid analytical geometry. 

347 



348 APPLICATION TO SURFACES. [Ch. XXXII. 

coordinate axes Ox, Or, Oz. Then, whatever be the numbers ?n and «, 
the point whose coordinates are 

y — ^ i z~> z — 



m -j- n m 4- n m -\- n 

is a j>oint on the straight line through the points x l9 y lt z l and x 2 ,y 2 , z 2 , 
which divides the segment between these points in the ratio of m to n. 

By varying /;/ and n we can make x' ,y' , z' the coordinates of any 
point whatever on this straight line. But the point x, y' , z' must 
be on the surface (i), since, on substitution, these values satisfy (i). 
Therefore, whatever be the two points whose coordinates satisfy (i), 
the straight line through these points must lie wholly in the locus 
representing (i). This is Euclid's definition of a plane surface. 

The intercept of the plane on the axis Oz is —D/C. Therefore, 
when C = o, the intercept is oo , or the plane is parallel to Oz. 
Hence (i) becomes 

Ax + By + D — o, 

the equation of a plane parallel to Oz, cutting the plane xOy in the 
straight line whose equations are z — o, Ax -\- By -\- I) = o. 

We use orthogonal coordinates unless otherwise specially men- 
tioned. If /, m, n are the direction cosines of the perpendicular from 
the origin on the plane and p is the length of that perpendicular, the 
equation of the plane can be written in the useful form 

Ix 4- my 4- nz — p = o, (2) 

where Z 2 4" nj2 + n% — J • 

II. The Straight Line. Since the intersection of any two planes is 
a straight line, the equations of a straight line are the simultaneous 
equations 

V + ^'+0 + A = o. ) K6) 

The equations (3) of a straight line can always be transformed 
into the symmetrical form 

x-a __y-b _ z- c 

-r - -5- - ~r ~ *' (4) 

where a, b, c is a point on the line ; /, m, n, the direction cosines of 
the line; and A is the distance between the points x, y, z and a, b, c. 

III. The Cylinder. A cylinder is any surface which is generated 
by a straight line moving always parallel to a fixed straight line and 
intersecting a given curve. The moving straight line is called the 
clement or generator, and the fixed curve the directrix of the cylinder. 

With reference to space of three dimensions and rectangular coor- 
dinates, any equation 

/(•v-.r) = o (5) 

is the equation of a cylinder generated by a straight line moving 



Art. 233.] STUDY OF THE FORM OF A SURFACE AT A POINT. 349 

parallel to Oz and intersecting the plane xOy in the curve /"(.v, r) = o. 
Yox f{x,y) = o is nothing more than the equation 

/(•*, j; *) = o 

in three variables, in which the coefficients of z are zero, and which is 
therefore satisfied by any x, y on the curve fix, y) =0 in xOy and 
any finite value of z whatever. 

In like manner f(y, z) = o, f[x, z) = o are cylinders parallel to 
the Ox, Oy axes respectively. 

IV. The Cone. A cone is a surface generated by a straight line 
passing through a fixed point, called the vertex, and moving according 
to any given law, such as intersecting a given curve called the directrix 
or base of the cone. 

Any homogeneous equation of the «th degree in x, y, z, such as 

F(x,y, z) = o, (6) 

is the equation of a cone having the origin as vertex. 

Let a, ft, y be any values of x, y, z satisfying (6). Then, since 
(6) is homogeneous, ka, kft, ky will also satisfy (6), and we shall have 

F(kx, ky, kz) = k n F(x,y, z) = o 

whatever be the assigned number k. The coordinates of any point 
whatever on the straight line through the origin and a, ft, y can be 
represented hy ka, k(i, ky. Therefore all points of this straight line 
satisfy (6). When the point a, ft, y describes any curve, the straight 
line through O and a, ft, y generates a surface whose equation is (6), 
and this is by definition a cone. 

If we translate the axes to the new origin — a, — b, — c, by writ- 
ing x — a, y — b, z — c, for x, y, z in (6), we have 

F{x — a,y — b, z — c) = o, (7) 

a homogeneous equation in x — a, y — b, z — c, which is the equa- 
tion of a cone whose vertex is a, b, c. 

232. General Definition of a Surface. — If F(x,y, z) is a continu- 
ous function of the independent variables x,y, z, and is partially dif- 
ferentiate with respect to these variables, we shall define the assemblage 
of points whose coordinates x, y, z satisfy 

F(x,y, z) = o (1) 

as a surface, and call (1) the equation of the surface. 

233. The General Equation of a Surface. — Let F(x,y,z) =z o 
be the equation of any surface. 

Then, by the law of the mean, we can write 

F(x,y, z) = F{x',y, z') 



35© APPLICATION TO SURFACES. [Cir. XXXII- 

in which the summation can be stopped at any term we choose, provided 
we write £, ?/, Z instead of x f , y', z' in the last term, where £, 7/, Z 
is a point on the straight line between x, y, sand x^y', z' . We can 
therefore always write the equation to any surface in the standard 
form 

+ Zt\ \ ( ' v - x "> h + o -->'') 1+ (« ~ *') i } '*= - (0 

This enables us to study the function as a rational integral function 
of x, y, z. 

If the equation of the surface be given in the explicit form 
z = f(x,y), then in like manner, by the law of the mean, we have 
for the equation to the surface 

. =A*',y) +^p { (* - *o £ + (.-, -/) ±, } >, 

in which the summation stops at any term we choose, provided in the 
last term we write B, for x' and 77 for y'; £, ?/ being a point on the line 
joining x,y to x\ y'. 

234. Tangent Line to a Surface. — A tangent straight line to a 
surface at a point A on the surface is defined to be the limiting posi- 
tion of a secant straight line AB passing through a second point B 
on the surface, when B converges to A as a limit along a curve on 
the surface passing through A in a definite way. 

To find the condition that the straight line 

x — x' _y —y' _ z — z' _ 
I ~ m ~ 71 

shall be tangent to the surface F(x, y, z) = o. 

The equation of a surface in implicit form is, § 233, (1), 

F( x ',y, 0') + (.v - *') g + (y -jOgJ + (• - O a7\+* = °- ( 2 > 

Substitute IX, mX, nX for x — x' , y —J' r , 2 — z , from (1) in (2). 
We have the equation in A, 



o = F { ,;y, 0') + (/ g + m % + n g) A + R, 



(3) 



for determining distances from x',y', z' to the points in which (1) 
intersects the surface (2). \{ F{x' , y' , z') = o, or x',y', z' is on the 
surface, one root of (3) is o. If in addition 

7 dF dF dF 



Art. 235.] STUDY OF THE FORM OF A SURFACE AT A POINT. 351 

two values of A are o, or two points in which the secant (1) cuts the 
surface (2) coincide in x', y' , z' , and the line will be tangent to the 
surface at x' , y' , z ', and have the direction determined by /, w, ;/. 

Observe that in conditions (4) and P -+- m 2 -f- n 2 = 1 we have 
only two relations to be satisfied by the three numbers /, m, n, and 
therefore there are an indefinite number of tangent lines to a surface 
at a point .1', y', z' '. 

If the equation to the surface be in the explicit form z = f{x,y), 
or 

. = *+<*-*)& + ,-y)V + J l, (5 ) 

then, as before, the straight line (1) meets the surface (5) in x',y', z' 
when z = z' and other points whose distances from x',y', z' are the 
roots of the equation in A, 

ox oy J 

The condition of tangency is that a second point of intersection 
shall coincide with x',y' , z' , or 

, of df 

235. Tangent Plane to a Surface. — When the locus of the tan- 
gent lines at a point on a surface is a plane, that plane is called the 
tangent plane to the surface at that point. The point is called the 
point of contact. 

Tangent plane to F[x,y, z) = o at x',y', z f . 
The straight line 

x — x' _ y — y' _ z — z' 
I m n 

is tangent to the surface F= o at x',y', z' when F(x',y', z') — o and 
7 dF dF BF ,\ 

/ a7 + ra a j 7 + w a? = °- (2) 

If now at x', y', z' the derivatives 

dF dF dF 
dx" dy" b\z' 

are not all o, we obtain the locus of the tangent lines to F= o at 
x',y,z f by eliminating /, m, n between (1) and (2). Therefore this 
locus is 

(*-*og+u-y)f + (— 0^ = 0. (3) 

Equation (3) is of the first degree in x,y, z, and therefore is a 
plane tangent to F = o at x',y', z'. 



352 APPLICATION TO SURFACES. [Ch. XXXII. 

Tangent plane to z -=.f(x,y). 
Eliminating /, m, n between (i) and 

l & +m W~ n = °' (4) 

we have 

as the tangent plane to z — f^x',y'. 

236. Definition of an Ordinary Point on a Surface. — We have 
just seen that when at any point on a surface F — o the first partial 
derivatives, 

dF dF dF 

dx' gp dP 

are not all zero, the surface has at that point a unique determinate 
tangent plane. Such a point is called a point of ordinary position, or 
simply an ordinary point. 

On the contrary, if at x,y, z we have 

d x F = o, d y F = o, d z F = o, 

the point is called a singular point on the surface. We shall see 
presently that the surface does not have a unique determinate tangent 
plane at a singular point. 

EXAMPLES. 

1. Find the conditions that the tangent plane to z =/[x, y) shall be parallel to 
xOy. Ans. d x f = d y /= o. 

2. Find the conditions that the tangent plane to F(x, y, z) = o shall be hori- 
zontal. Ans. d x F = d y F = o, d z F ^ o. 

3. Show that the tangent plane at x', y', z to the sphere x 2 -f- y 1 -|- z 2 = a 1 is 

XX> + yy' 4- zz> - a 2 . 



x* , y 



c-2 



4. Find the tangent plane to the central conicoid 1 1 = 1. 

a b c 

xx' yy' zz' 

Ans. V ~-\ = I- 

a b c 

5. Show that the tangent plane to the paraboloid ax 2 -[- by % = iz at x' . y', z' 
is nxx' -j- byy' = z -f- z'. 

6. Show that the tangent plane to the cone Fx,y, z) = o, having the origin as 
vertex, is xd x 'F -f- yd y >F -\- zd z >F — o. 

This follows directly from the fact that F is homogeneous, and therefore the 
tangent plane is 

dF t dF t dF dF , dF dF 

x d X ' + y w + Z ^ = x te+ y w + * d~T = n ^ x '>y'> ■o = °- 

where n is the degree- of the cone. 



Art. 236.] STUDY OF THE FORM OF A SURFACE AT A POINT. 353 

7. Find the equation to the tangent plane at any point of the surface 
jr* 4- y* -f- z* = a*, and show that the sum of the squares of the intercepts on the 
axes made by the tangent plane is constant. 

8. Prove that the tetrahedron formed by the coordinate planes and any tan- 
gent plane to the surface xyz = a 3 is of constant volume. 

9. Show that the equation of the tangent plane to the conicoid 

ax' 1 -\- by- -f- cz 2 -f- 2fyz -j- 2gxz -f 2/ixy -\- 2nx -f- zvy + zwz -f d = o, 
at x J , y', z', is 

(ax' + hy' -f gz' 4 u)x + (Ax' -f by' + fz' + v)y + 
(^ +jy + «* + «0* + «y + z// -f wz' + fl? = o. 

10. Show that 



»[ 



-*>gl+< K -'>| +<*-*>£' 



F = o 



is the general equation of any conicoid, and that 

dF _ dF _ dF _ 

dx dy dz 

are the equations of the center of the surface. 

11. Show that the plane 

dF n dF dF 

( ,_ a) _ +0 ,_ / Q_ + ( ,_ r) _ =o 

cuts the conicoid F = o in a conic whose center is a, /?, y. and therefore this is 
the tangent plane when a, ft, y is on the surface. 

12. Show that the locus of the points of contact of all tangent planes to the 
surface F = o, which pass through a fixed point a, (5. y, is the intersection of 
F = o with the surface 

dF dF dF 

13. This surface is of degree n — 1 when F — o is of degree n. 

For, let F— U n -f- ... + U x -\- U . where U r is the homogeneous part of 
degree r. Then, as in two variables, we have the concomitant 

dU r , dU r dU r 

X Hx- + y ~dy- + Z -dz- = rU - 

Therefore the tangent plane at x, y, z may be written 

dF dF dF 

X dx~ JrY -dy- +Z -dz- = nUn + ( " ~ l)Un - 1 + ' * ' + U " OT 
dF dF dF 

x a7 + Y T + z a* + ^" r + 2Un ~ 2 + ■•• + (*- ^ + W ^o = o. 

since £4 -f~ . . . -f- U Q = o. 

14. Find the condition that the plane lx -f- 09/ -f »2 = o shall be tangent to 
the cone 

F = ax 2 -f ^/ 2 4- r2 2 4- 2/J/3 4- 2gxz 4- 2hxy = o 

at *', /, z'. 

The equation to the tangent plane at x', y', z' is, Ex. 6, 
dF dF dF 

X dS+Sd/^ Z d7=°' 



354 



APPLICATION TO SURFACES. 



[Ch. XXXII. 



In order that this shall be identical with ix -f- my -f- «* = o, the coefficients 
of x, J', z must be proportional. 

9^ / dF 



OF I . dF I dF I 



Hence 



ax' + hy' -j- #2/ = /A, 
//jf' -f- £/ -f-/z' = wA, 
^ + />'' + c* = nX, 
lx' -j- m )'' ~\- nz' = o. 
In order that these equations shall be consistent we have 

g I — o, 



* / 
/ < 



the required condition. 

15. Generalize Ex. 14 by finding the conditions that the plane may cut, be tan- 
gent to, or not cut the cone except in the vertex. 

Eliminating z, the horizontal projection of the intersection of the plane and cone 
is two straight lines 

(an 2 -j- d 2 — 2ghi)x 2 -f- 2(hn 2 -j- dm — gmn —fhi)xy -j- (lui 1 -j- cm 2 — 2fmn)y 2 = o. 

These will be real and different, coincident or imaginary, according as 

(an 2 -f- cP — 2ghi)(bri 2 -\- cm 1 — 2fmn) — (Jin 2 -j- elm — g??m — f/n) 2 , 

which can be written as the determinant 

A = 



a h 


g 


I 


h b 


f 


m 


8 f 


c 


n 


I m 


11 


O 



is negative, zero, or positive, respectively. 

16. Show that the projections of the two lines in 15 can be written 

dj dzl dA , 

W X --0~h X} ' + Ta y -=°' 
with similar equations for the projections on the other two coordinate planes. 

17. Show that A in Ex. 15 can be written 



da do 



dc 



where 



, 3D 

d g 

D 



3D 

l ~zr-~ -J- 



¥ 



lm M = ~ A > 



a 


k 


g 


h 


b 


f 


g 


f 


c 



237. Conventional Abbreviations for the Partial Derivatives. — 

The elementary study of a surface is usually confined to those properties 
which depend only on the first and second derivatives, that is, on the 
quadratic part of the equation to the surface when the equation is 
expressed by the law of the mean. 

This being the case, it is of great convenience in printing and 
writing to have compact symbols for the first and second partial deriv- 
atives. These derivatives being the coefficients of the first and second 
powers of x,y, z in the equation, it is customary to represent them by 



Art. 238.] STUDY OF THE FORM OF A SURFACE AT A POINT. 355 

the same letters as are conventionally employed as the coefficients of 
the terms in the general equation of the second degree in three vari- 
ables. 

We shall hereafter frequently write : 

When F(x, v, z) = o, 

"£ '•%• »■'£■ 



d*F = d*F d 2 F 



Fm ™ '■Q*™ ff m * F 



dy dz dx dz dx dy 

When z = /{x,y), 

* = d Z = d Z = d V = d Y , = d V 

P dx' q dy ' T ~ dx*' * ~ dxdy' df' 

238. Inflexional Tangents at an Ordinary Point. — We have seen, 
§§ 234, 235, that there are an indefinite number of tangent lines to a 
surface at an ordinary point, lying in the tangent plane and passing 
through the point of contact. If the second partial derivatives of 
F = o are not all o, there are two of these tangent lines that are of 
particular interest. 

(1). Let z =f(x,jy) be the equation of a surface. 

The straight line 

I m n ' 

cuts the surface z = _/"in points whose distances from the point x, y, z 
on the surface are the roots of the equation in A 

-(^H-g-.) + j(/i + ^)/ + >=.i, w 

we have seen that (1) is tangent to z = / 'at x, y, z. 

If in addition we have /, m, n satisfying the condition 

two roots of (2) are o, and the line (1) cuts the surface in three coin- 
cident points at x, y, z. 
The conditions 

pi -\- qm — n = o, 
rP -j- 2slm -f- lm 2 = o, 



35 6 APPLICATION TO SURFACES. [Ch. XXXII. 

determine two straight lines, in the tangent plane, tangent to the 
surfaces =/"at the point of contact. Each cuts the surface in three 
coincident points there. 

These are called the inflexional tangents at x, y, z. They are real 
and distinct, coincident, or imaginary, according as the quadratic 
condition 

rl 2 -{- 2s/m -f- trri 1 = o, 

in l/m, has real and different, double, or imaginary roots, or accord- 
ing as 



rt — s 2 = 



_ &f &f 



dx 2 dy 



\dxdy) 



is negative, zero, or positive. 

Since any straight line, such as (i), cuts any surface of the »th 
degree in n points, the straight lines in any plane cut the curve of sec- 
tion of a surface of degree n in n points. Therefore a plane cuts a sur- 
face of the nth degree in a plane curve of degree n. 

The tangent plane to a surface of degree n cuts the surface in a 
curve of degree n passing through the point of contact. But each of 
the inflexional tangents to the surface cuts this curve in three coincident 
points at the point of contact. Each is therefore tangent to the curve 
of section at the point of contact of the tangent plane, which is there- 
fore a singular point on the curve of section. This point is a node, 
conjugate point, or cusp according to the value of condition (5). Com- 
pare singular points, plane curves. 

Eliminating l,m,n between (1), (3), (4), we have for the equa- 
tions of the inflexional tangents at x, y, z 

Z-z=(X-x)p + (V-y)q, 

{X - xfr + 2 (X - x){F-y)s + {V-yft = o. 

The second is the equation of two vertical planes cutting the first, 
the tangent plane, in the inflexional tangents. 

(2). If the equation of the surface is F— o, then the straight line 
(j) cuts the surface in points whose distances fvomx,y, z are the roots 
of the equation in A, 

^ ,/dF dF dF\ X 2 / d d a \a^ , „ 

If x,y, z is on the surface, or F(x,y, z) = o, and 

LI + Mm +Nn == o, 

the line (1) is tangent at x,y, z. If in addition /, m, n satisfy the 
condition 



{ / 3x+ m 6y+ n dz) F =°> 



Art. 239.] STUDY OF THE FORM OF A SURFACE AT A POINT. 357 

the line (1) cuts the surface in three coincident points at x,y, z. The 
conditions 

P + m 2 _|_ f? s- lf (6) 

LI -f- Mm -f Kn = o, (7) 

^Z 2 + Bm 2 -f C* 2 + 2*>*» -f 2G/» + 2^/w = o, (8) 

determine the directions of the two inflexional tangents. 

Eliminating /, m, n between (1), (7), (8), we have the equations of 
the inflexional tangents a.tx,y, z, 

{<*-* )-& + (*-*#+ (*—)£ } /■= o, ( 9 ) 

{(^- J; )^+(i'-^)| + (Z- e )l}V=o. (xo) 

The first is the tangent plane, which cuts the second, a cone of the 
second degree with vertex x,y, z, in the two inflexional tangents. 

These tangents will be real and different, coincident, or imaginary, 
according as the plane (9) cuts the cone (10), is tangent to it, or 
passes through the vertex without cutting it elsewhere. That is, ac- 
cording as the determinant (see Ex. 15, § 236) 

A H G L 

H B F M (11) 

G F C N 
L M No 

is negative, zero, or positive. 

239. Should the second partial derivatives also be separately o at 
x, v, z, and r the order of the first partial derivatives thereafter which 
do not all vanish at x, y, z, then there will be at x, y, z on the sur- 
face r inflexional tangents, which are the r straight lines in which the 
tangent plane at x, y, z cuts the r planes 

or the cone of the rth degree, 

{(X- x) l x + ( r- y) d - + ( z- Z ) £}> = <». 

These r inflexional tangents to the surface are the r tangents to 
the curve cut out of the surface by the tangent plane at the point of 
contact, which point is an r-ple singular point on the curve of 
section. 

EXAMPLES. 

1. Show that the inflexional tangents at any point x' , /, 2' on the hyperboloid 
x^/a 1 4. y 2 /b 2 — z 2 /c* = 1, lie wholly on the surface and are therefore the two 
right-line generators passing through the point. Show that their equations are 



y - y 



bx'z' ± acy' ay' z' q= bcx 



35 8 APPLICATION TO SURFACES. [Ch. XXXII. 

2. Show that the inflexional tangents at a point x, y, z on the "hyperbolic parab- 
oloid x 2 /a 2 — y 2 /b 2 = 2s lie wholly on the surface, and that their equations 
are 

X - x _ Y-y _ Z -z 
a ~ ± b ~ x y 
a T I 
the upper signs going together and the lower together. 

3. Show that the inflexional tangents to the cone 

ax 2 -f by 2 -\- cz 2 -j- ifyz -J- 2gxz -f 2/kr_y = O 
are coincident with the generator through the point of contact. 

4. Show that at a point on a surface at which any one of the coordinates is a 
maximum or a minimum the inflexional tangents are imaginary. 

240. The Normal to a Surface at an Ordinary Point. — The 

straight line perpendicular to the tangent plane at the point of contact 
is called the normal to the surface at that point. 

Since the equation to the tangent plane at x, y, z is 

the coefficients of X, Y, Z are proportional to the direction cosines 
of the normal, and we have for the equation to the normal at x, y, z 



or 



EXAMPLES. 

1. Show that the normal at x, y, z to xyz = a z is 

Xx - x 2 = Yy - y 2 = Zz - z 2 . 

2. Find the equations of the normal to the central conicoid ax 2 -f- by 1 + &* = *• 

X - x _ Y-y _ Z- z 
ax ~ by ~ cz 

3. Show that the normal to the paraboloid ax' 1 -f- by 2 — 22 has for its equations 

X - x Y - v 




by 



Z. 



241. Study of the Form of a Surface at an Ordinary Point. 
— We may study the form of a surface at an ordinary point by 
examining it (1) with respect to the tangent plane, (2) with respect to 
the conicoid of curvature, (3) with respect to the plane sections parallel 






Art. 242.] STUDY OF THE FORM OF A SURFACE AT A POINT, 359 

to the tangent plane, (4) with respect to the plane sections through the 
normal. 

242. With respect to the tangent plane: 

(1). Let z —/(x,v). Then the equation of the surface is 

Let A", J^, ^ be a point in the tangent plane in the neighborhood 
of the point of contact x, y, z. Then the difference between the 
ordinate to the surface and the ordinate to the tangent plane is 

z-z I= i{(A--.v ) i + (^-, ) ;;;>. 

This difference is positive for all values of X, Y in the neighbor- 
hood of x, y when 

dyay /a 2 / y dy 

S? aF ~~ \dx~dy) an a? 

are positive (Ex. 19, §25). Then in the neighborhood of the point 
of contact the surface lies wholly above the tangent plane, and is said 
to be convex there. 

In like manner Z — Z is negative throughout the neighborhood 
when rt — s 2 is positive and r is negative at the point of contact. 
Then the surface in the neighborhood of the point of contact lies 
wholly below the tangent plane and is said to be concave there. 

(2). Let F(x, y, z) = o. In the same way we have the equa- 
tion of the surface, 

(X-x)F' x +(Y-y)F' y +(Z-z)F', = 

and for that of the tangent plane at x, y, z, 

(X - x)F' x + {V-j)F;+ {Z 1 - z)F' z = o. 
On subtraction, 

(z^F'-l j ( *_^ +( r-^+(ir-*)± } V. 

Therefore, at x,y, z, by Ex. 20, § 25, when 



\A H\ and A 

\H B\ 



A H G 
H B F 
G F C 



are positive, the surface is convex when A and F' z are unlike signed, 
concave when A and F' are like signed. 



360 APPLICATION TO SURFACES. [Ch. XXXI.. 

Observe that a surface is concave or convex at a point when tie 
inflexional tangents there are imaginary, and conversely. Wheu a 
surface is either concave or convex at a point, its form is said to «e 
synclastic there. When the inflexional tangents are real and different 
the surface does not lie wholly on one side of the tangent plane in the 
neighborhood of the point of contact, but cuts the tangent plane in a 
curve having a node at the point of contact and tangent to the inflex- 
ional tangents. At such a point the form of the surface is said to be 
anticlastic, and the surface lies partly on one side and partly on the 
other side of the tangent plane in the neighborhood of the contact. 

The conditions that a surface may be synclastic or anticlastic at a 
point are, (n), § 238, 



A H G L 

H B F M 

G F C N 

L M N o 



-}- synclastic y 
— anticlastic. 



The hyperboloid of one sheet and the hyperbolic paraboloid are the 
simplest examples of anticlastic surfaces, these being anticlastic at 
every point of the surfaces. The surface generated by the revolution 
of a circle about an external axis in its plane generates a torus. This 
surface is anticlastic or synclastic at a point according as the point is 
nearer or further from the axis of revolution than the center of the 
circle. 

243. With Respect to the Conicoid of Curvature. 

(1). The explicit equation z = /(x,y), or 

t = z+(x - x) V +{ r- y) !+I { { x- x) l s+( r- y) £ J 7, 

shows that in the neighborhood of x,jy, z the surface differs arbitra- 
rily little from the paraboloid 

z = z+{ x- x) d J- +( y- y) ^ + 1 { { x- x) ±- +{ r- y) I } '/ 

This is called the paraboloid of curvature of the surface at x,jy, z. It 
has the same first and second derivatives at x, y, z as has the surface 
z —f, and therefore, at that point, has, in common with the surface, 
all those properties which are dependent on these derivatives. 

Obviously, the surface is synclastic or anticlastic according as the 
paraboloid is elliptic or hyperbolic. 

From analytical geometry, the discriminating quadratic of the 
paraboloid 

rx 2 -f- ty 2 -j- 2sxy -f- 2px -\- 2ay — 20 -\- k = o 
is X 2 - (r + t)X + (rt — s 2 ) = o. 



Art. 245.] STUDY OF THE FORM OF A SURFACE AT A POINT. 361 

This gives the elliptic or hyperbolic form according as rt — s 2 is 
positive or negative. 

(2). In the same way, the implicit equation F(x,y, z) — o, or 



dx 



+ 



&-*)&+ l r -')l + '*-')sV*=*' 



shows that in the neighborhood of x,y, z the surface differs arbitrarily 
little from the conicoid of curvature whose equation is the same as the 
left member of the equation above when equated to o. The form 
of the surface at x, y, z is the same as that of the conicoid of curvature 
there, and they have the same properties there as far as these proper- 
ties are dependent on the first and second derivatives of F. 

The discrimination of the conicoid can be made through the 
discriminating cubic (see Ex. 17, p. 30) 

A — A, H , G | = o, 

H , B -A, F 

G , F , C-X\ 

and the four determinants 



A 


H G L\ 


H 


B 


F Ml 


G 


F 


C N\ 



as in analytical geometry.* 

244. The Indicatrix of a Surface. — At an ordinary point x,y, z 
on a surface, at which the second derivatives are not all o, a section 
of the surface by a plane parallel to and arbitrarily near the tangent 
plane differs arbitrarily little from the section of the conicoid of 
curvature made by this plane. Such a plane section of the conicoid 
of curvature is called the indicatrix of the surface at x, y, z. 

Points on a surface are said to be circular (umbilic), elliptic, para- 
bolic, or hyperbolic according as the indicatrix is a circle, ellipse, 
parabola (two parallel lines), or hyperbola (two cutting lines). 

245. Equation to Surface when the Tangent Plane and Normal 
are the 2-plane and 2-axis. — If the equation is z =f(x, y), then since 
z — o, p = o, q = o at the origin, the equation is 

2Z = rx 2 -f- 2sxy -\- ly 2 -f- 2R. 

The equation of the indicatrix at the origin is 

z = rx 2 -j- 2sxy -\- ly 2 , 



* See Frost's, Charles Smith's, or Salmon's Analytical Geometry. 



3 62 



APPLICATION TO SURFACES. 



[Ch. XXXII. 



z being an arbitrarily small constant. This is an ellipse or hyperbola 
according as rt — s 2 is positive or negative, giving the synclastic or 
anticlastic form of the surface there accordingly. 

246. Singular Points on Surfaces. — If, at a point x, y, z on a 

surface F ■=. o, we have independently 

dF OF dF 

a — = °> a - — °> a~~ 
ox ay az 

the point is said to be a singular point. 

If the second derivatives are not all zero, then all the straight lines 
whose direction cosines /, ??i, ?i satisfy the relation 

d 3 \2 



(1) 



4+ 



dy 



dz 



F = 



(2) 



will cut the surface in three coincident points at x,y, z, and are called 
Eliminating /, m, n by means of the equation to the 



tangent lines. 



line and (2), we obtain the locus of the tangent lines at x,y, 

{ { x-x)± + { r 



S)j^ + ( Z -z) 



% 



a 

~dz 



F= o. 



(3) 



This is the equation of a cone of the second degree, with vertex 
x, y } z, which is tangent to the surface /=o at the point x, y, z. 
The form of the surface at x, y, z is therefore the same as that of this 
cone. Such a point is called a conical point on the surface. 

When this cone degenerates into two planes, then all the tangent 
lines to the surface at x, y, z lie in one or the other of two planes. 
The point is then called a fiodal point. The condition for a nodal 
point is that equation (3) shall break up into two linear factors, or 



A 

H 

G 



H 
B 
F 



G\-. 

F 

C 



(4) 



A line on the surface i^^oat all points of which (4) is satisfied 
is called a ?wdal line on the surface. Such a line is geometrically 
defined by the surface folding over and cutting itself in a nodal line, 
in the same way that a curve cuts itself in a nodal point. 

If r is the order of the first partial derivatives which are not 
all zero, then the surface has a conical point at x,y, z of order r, and 
a tangent cone there of the rth degree whose equation is 



(X 



6 
dx 



C->)|r + ^— )w} >=0 - (5) 



247. A singular tangent plane is a plane which is tangent to a 
surface all along a line on the surface. For example, a torus laid on 
a plane is tangent to it all along a circle. The torus has two singular 



Art. 247-] STUDY OF THE FORM OF A SURFACE AT A POINT. 363 

tangent planes. All planes tangent to a cylinder or cone are 
singular. 

EXERCISES. 

1. The tangent plane to yx 2 = a 2 z at x lt y v z x is 

zxx x y x -J- y x\ — a 2 Z = 2(/'': r 
Find the equation to the normal there. 

2. The tangent plane to z(x 2 -\- y 2 ) = 2kxy at x lt y x1 z t is 

2x(x l z l - ky x ) + 2y(y 1 z 1 -kx 1 ) -f z(x\ -f y\) - 2kx x y x = o. 

The tangent plane meets the surface in a straight line, and an ellipse whose 
projection on the xOy plane is the circle 

(x 2 + y 2 ){x\ - y\) + {x\ + yi){yy x - xx x ) = o. 

Show that the s-axis is a nodal line. 

3. The tangent plane to a 2 y 2 — x 2 (c 2 — z 2 ) at x v y lt z x is 

XXl (f - z \) - a 2 yy x - zz x x\ + x\z\ = O. 

At any point on Oz, F x ' = F y ' =± F z ' = o, show that at any such point there 
are two tangent planes 



z_ ± \t=£ 



4. Show that the tangent plane at x x , y x , z x to 

x $ _j_ jj/3 _j_ 2 3 — $xyz = a z 
is x{x\ - y x z x ) -f y{y\ - x x z x ) + z{z\ - x x y x ) = a\ 

5. The tangent plane at x x , y x , z x to x m y n zP = a is 

;;/ n p 

— x -\ y -f- — r z = m -\- n -\- p. 

x x y x z 

6. Show that (2a, 2a, 2a) is a conical point on 

xyz — a(x 2 -\- y 2 -f- z 2 ) -\- 4a 3 = o, 

and find the tangent cone at the point. 

Aits, x 2 -\- y 2 -\- z 2 — 2yz — 2zx — 2xy — o 

7. Show that the surface 

\a 2 ^ b 2 + c 2 ) 6 \a 2 ^ b 2 ) c 2 ^ 4 

has two conical points. 

The tangent cone at o, o, o is $x 2 /a 2 -\- 3y 2 /b 2 -f- z 2 /c 2 = o. 

8. Determine the nature of the surface 

ay* + bz 2 + x(x 2 + y 2 + z 2 ) = O 
at the origin. 

The origin is a singular point, the tangent cone there is ay 2 -j- bz 2 = o. If a 
and b are like signed, the origin is a cuspal point around the x-axis. 

9. A surface is generated by the revolution of a parabola z 2 = <p?ix about an 
ordinate through the focus; find the nature of the points where it meets the axis of 
revolution. 



364 APPLICATION TO SURFACES. [Ch. XXXII. 

Hint. The equation of the surface can be written 

The two right-angled circular cones x 2 -f- y 2 = (z ± itrif are tangent to 
the surface at the singular points. 

10. If tangent planes are drawn at every point of the surface 

a(yz -f z x + xy) = xyz, 

where it is cut by a sphere whose center is the origin, show that the sum of the 
intercepts on the axes will be constant. 

11. Show that the general equation of surfaces of revolution having Oz for axis 

x 2 +y 2 =/{z). 

Thence show that the normal to the surface at any point intersects the axis of 
revolution. 

12. Show that at all points of the line which separates the synclastic from the 
anticlastic parts of a surface the inflexional tangents must coincide. 

13. The equation of an anchor-ring or torus is 

Show that the tangent plane at x', y' , z', is 

(r - tyxx' +yy') + rzz' = r\a 2 -f c(r - c)], 
where r 2 = x' 2 -\- y' 2 . 

The tangent plane at any point on the circle x 2 -\-y 2 = (c — a) 2 cuts the surface 
in a figure 8 curve whose form is given by the equation 

(y 2 + z 2 ) 2 — Aacy 2 + \c(c — a)z* = o. 

14. When the tangent plane passes through the origin it cuts two circles out of 
the torus which intersect in the two points of contact. 

15. Show that the cylinder x 2 -\- y 2 ■=. c 2 cuts the torus in two parts, one of which 
is synclastic, the other is anticlastic. 



CHAPTER XXXIII. 



CURVATURE OF SURFACES. 

248. Normal Sections. Radius of Curvature.— The normal 
section of a surface at a point is the curve cut on the surface by a 
plane passing through the normal to the surface at the point. 

To find the radius of curvature of a normal section. 

Let the tangent plane and normal at 
an ordinary point on the surface be taken 
as the 2-plane and 2-axis respectively. 
Then the equation to the surface can be 
written 

z = \{rx~ + 2sxy + tf) + J?, (1) 

since at the origin z = o, p = o, q = o. 

Cut the surface by a plane passing 

through Oz and making an angle 6 with 

Ox. At every point of this plane let 

x — p cos 6, y — p sin 6. FlG - *42. 

. \ z = \p 2 {r cos 2 6 + 2s cos 6 sin -f / sin 2 ^) + T, 

where T contains as a factor a higher power of p than p l . 

The radius of curvature R of this normal section PO is, by New- 
ton's method, §101, Ex. 4, given by 




I _ f*2Z 



(2) 



(3) 



= r cos 2 6 -f- 2s cos 6 sin 6 -J- / sin 2 6^, 
— 1 ( r + /) -f i(r — /) cos 2 -f- s sin 2 #. 

The directions of the normal sections in which the radius of cur- 
vature is a maximum or a minimum are given by the equation 

tan 2d = y— t . (4) 

If a is the least positive value of satisfying (4), the general solu- 
tion is \nn -f- a, showing that the normal sections of maximum and 
minimum curvature are at right angles. These sections are called the 
principal sections of the surface at the point considered. Their radii 
of curvature at the point are called the principal radii of curvature. 

565 



3 66 



APPLICATION TO SURFACES. 



[Ch. XXXIII. 



If the principal sections be taken for the planes xOz,yOz, the ex 
pression for the radius of curvature of any section will be 

■ = r cos 2 tf + / sin 2 #, 



R ' ' (5) 

since then s = o, by (4). 

Let R x and R 2 be the radii of the principal sections. 
Then when 6 = o; R- 1 = r; 6 = \n, R~ l = /, in (5). 
1 cos 2 ^ sin 2 # 

Also, if R' is the radius of curvature of a normal section perpen- 
dicular to that of R, then 

1 _ sin 2 # cos 2 # 

~r' ~~ ~rt + "^r* 



R ^ K' R, 



+ *■ 



(6) 

(7) 



The sum of the reciprocals of the radii of curvature of normal 
sections at right angles is constant. This is Euler's Theorem. 

249. Meimier's Theorem. — To find the relation between the 

radii of curvature of a normal section 
and an oblique section passing through 
the same tangent line. 

Take xOz as the normal plane, and 
let the oblique plane xPOQ make the 
angle with xOz. 

Then the equation of the surface is 
2z = rx' 1 -\- 2sxy -j- (v 2 




d 



Fig. 143. 



3 \ °s J °n 

At any point P in the oblique sec 
tion y = 3 tan 0. 



/• 



2Z 
~^2 



z z 2 , x I d 

r 4- 2 j — tan + / ., tan-0 4— ;.— 
1 x X- 3 \d£ 

But since Ox is tangent to the curve OP at o, 

z sec 0_ /*s 



z a 

- tan — - 
x 011 



/■ 



£ 



x 



JL x 



£ 



x- ' [Ox 2 , 

as P converges to O along P.O. Also, in the xOz section, if 
MR = x , we have j' = o, and 



£z - m.. 



Art. 250.] 



CURVATURE OF SURFACES. 



367 



Let R , R be the radii of the normal and oblique sections. Then, 
for s(=)o, 



R 



f- 

J j 2Z 



COS 0, R. 



JL 2z ' 



Hence 



R = R Q cos 0. 
This is Meunier's theorem. 

250. Observe, in the equation to the surface (1), § 248, the equa- 
tion of the indicatrix is 

2Z = rx 2 -f- 2sxy -f if. (1) 

The principal sections of the surface at pass through the axes 
of the indicatrix conic, whose equation is 

2Z - roc* + if ( 2 ) 

when xOz and yOz are the principal planes. 

Equation (1) shows that the radius of curvature of a normal section 
varies as the square of the corresponding central radius vector of the 
indicatrix. All the theorems in central conies which can be expressed 
by homogeneous equations in terms of the radii and axes furnish 
corresponding theorems in curvature of surfaces. 

We shall adopt the convention that the radius of curvature of a 
normal section of a surface is positive or negative according as the 
center of curvature of the section is above or below the tangent 
plane. 

When the indicatrix is an ellipse the principal radii have like 
signs, and have opposite signs when the indicatrix is the hyperbola. 
The inflexional tangents are the asymptotes of the indicatrix. 

251. At any point of a surface to find the radius of curvature of a 
normal section passing through a given tangent line at the point. 

Let F= o be the equation of the surface. Let P be the given 
point x,y, z, and /, m, ?i the direction cosines of the tangent line 
there. Let Q be another point X, Y, Z on the surface and in the 
normal section. 

Let QR be the perpendicular from Q on the tan- 
gent line PR. 

Then for R, the radius of curvature of the sec- 
tion, we have 



R: 



I 



£ PQ* (PR 



-£ 



PQ 1 
2QR 



PR 2 _ 

2~QR~X ^QR\PQ 
The tangent plane at P is 
(X-x)L + {Y-y)M+ (Z-z)N 
The distance of Q from this plane is 

QR = i- < (X -x)L + (r-y)M+ (Z - z)A'[, 

K 

rhere k> = Z 2 + M 2 + .V 2 . 




144. 



3 68 



APPLICATION TO SURFACES. 



[Ch. XXXIII. 



Also, Q being a point on the surface, 
(X- x)L + (V-y)M+ (Z- z)N 



■ i=I 



^L+v-A^+iz 



dx 



dz\ 



F+ 2T 



: AP + BnP -f Cn % -f 2 Fmn + 2 Gln + zHlm, 
since £T = o for Q( = )J>, and 

X - x _ Y-y _ Z-z _ 



(i) 



is the equation of the tangent PR. The derivatives Z, A, etc. , of 
course being taken at P. 

252. If the equation of the surface be f(x,y) — z = F — o, 
then since L ~p, M = q, N= — 1, C = F — G = o, (1), § 251, 
becomes 

1 rP 4- 2slm 4- ^ 2 

(1) 



K 



1/i +/ 2 + ^ 



253. To Find the Principal Radii at Any Point on a Surface. 

— We have only to find the maximum and minimum values of R in 
(!)> § 2 5 J > § 2 52. 

I. In (1), § 251, let /, m, n vary subject to the two conditions 

IL + mM+nN =0, P + w 2 -f ?z 2 = 1. 
Then, by the method of § 217, 

Al + Zfa + £« + AZ + fjil = o, 
#7 -f- Bm + Fn + AJ/-f jura = o, 
GI + Zw + Cn +\N -f yuw = o. 
Multiply by /, m, n, respectively and add. . *. jx = — k/R. 
... (A - k/R)1 + #« + G« + AZ = o, 

HI + (B — K/R)m -f Z« -f \M— o, 

G7 + Zrc +(C-/c/Z)«+ AiV^= o, 

Z/ -f j&/>w + Nn = o. 

Eliminating /, m, n, A, we get the quadratic 

A -k/R, H , G , Z 

, B - k/R, F 



H 
G 
L 



, M 



, M 
C - k/R, N 

N ' , o 



Art. 254.] CURVATURE OF SURFACES. 369 

the roots of which are the principal radii of curvature at the point at 
which the derivatives are taken. 

II. If z = f(x,y) be the equation to the surface, then in (1), 
§252, we have /, m, n subject to the two conditions 

pi -\- qm — 11 — o, and 1 2 -\- m 2 -f- ?i 2 = 1, 
which reduce to the single condition 

(1 +/»)/» + ipqlm -f (1 + f)m 2 = 1. 
Applying the general method for finding the maximum and 
minimum values to (1), § 252, 

rl + sm + A[(i + f)l + 'pqm\ = o, 
si + Im + \\_pql + (1 + q-)m\ r= o. 

Multiply respectively by / and m and add. Whence A = — k/R. 
Eliminating / and m from 

\rR - (1 + p 2 )*]l + (sR - pq K )m = o, 
(sR - p qK )l -f \iR — (1 -f- f)K\m = o, 
there results the quadratic 

[rR - (1 +J)k] \iR - (1 + ?>] - 0* - ^/c) 2 = o, 
or 

(rt - s 2 )R 2 - [r(i + ? 2 ) + /(i + f) - 2pqs-] K R + /c 4 = o, 

for finding the radii of principal curvature. In this equation 

*•=!+/» + ,». 

The problem of finding the directions of the principal sections and 
the magnitude of the principal radii of curvature is the same as that of 
finding the direction and magnitude of the principal axes of a section 
of the conicoid 

Ax 2 + By 2 + Cz 2 -f 2Fyz -f 2 G x z + *Hxy = 1, 
made by the plane Lx -j- i?/y -f- A% = o. 

254. To Determine the Umbilics on a Surface. — At an umbilic 

the radius of normal curvature is the same for all normal sections. 
Consequently equation (1), § 251, for any three particular tangent 
lines will furnish the conditions which must exist at an umbilic. 

Through any umbilic pass three planes parallel to the coordinate 
planes cutting the tangent plane there in three tangent lines whose 
direction cosines are l x , m lf o; / 2 , o, « 2 ; o, m z , ;/ 3 , respectively. Then 
equating the corresponding values of k/R in (1), § 251, 

AQ+Bm* + iBfa = Al? + Cn? + aGfo = Bmf+Cnf+zFm^. 

Also, since these three tangent lines are parallel to the tangent 
plane, the equations 

Z/j + Mm x — ZI 2 -f Nn 2 = Mm 3 + Nn z = o 
give 

1 ~~ Z 2 4- M» x L 2 + M 2 ' 



37° APPLICATION TO SURFACES. [Ch. XXXIII. 

and I lf m l have opposite signs. The same equations give like values 
for / 2 , n 2 , etc. On substitution we obtain the conditions which must 
exist at an umbilic, 

AM 2 + BL 2 - 2HLM _ AN 2 + CL 2 - iGLN 



L 2 4- M 2 I 2 + N 2 

BN 2 -f CM* - 2FMN 



W 



M » + A' 2 

These two equations in x,jy, z, together with the equation to the 
surface, give the points at which umbilics occur. 

If the equation of the surface isf(x,y) — z = o, results are cor- 
respondingly simplified and the conditions which must exist at an 
umbilic are immediately obtained from the fact that k/R is constant 
for all values of /, m, n, satisfying the identical equations 

— = rl 2 -j- 2slm -f- tm 2 , 

R 

I = (i +p 2 )P + 2pqlm + (i + q*)m\ 
Whence results, from proportionality of the constants, 
K r s t 



R i+p 2 pq i-\-q 



2* 



(0 



255. Equations (2), § 254, are very simply obtained by seeking 
the point on the surface z = f(x,j>) at which the sphere 

<P(x,y, *) = (x - a) 2 + {y - 0) 2 + (z - y) 2 - p 2 - o 

osculates the surface z —f. The first and second partial derivatives 
of z in are the same as those for/* at the point of osculation. Dif- 
ferentiating (p = o partially with respect to x and_>% we get 

x — a-f (z — y)p— o, 

y - P + (* - r) ? = °> 
1 +? + <* — y)r = °> 

1 + ? 2 + (* — r) / = °> 
pq + - r) s = °- 

, v /? I + P 2 I + ^ 



Also, ^ = — (2 — x)|/i + /~ + q 2 , since the direction secant of 
the normal with the £-axis is — (1 -\- p 2 -f- q 2 )*' 

256. Measure of Curvature of a Surface. — The measure of cur- 
vature of a surface is an extension of the measure of curvature of a 
curve in a plane, as follows: 






Art. 256. J CURVATURE OF SURFACES. 371 

The measure of entire curvature of a curve in a plane is the amount 
of bending. Let P x and P % be two points on a 
curve whose distances, measured along: the curve, 
from a fixed point are s x and s 2 . Let cp x and c\ be 
the angles which the tangents at P lt P 2 make with 
a fixed line in the plane of the curve. Then the 
whole change of direction of the curve between P 
and P 2 is the angle 2 — X . This angle is also 
the angle through which the normal has turned as a 
point P passes from P x to P 2 along the curve. 

This angle between the normals is called the entire curvature of 
the curve for the portion P x P r It can also be measured on a standard 
circle of radius r, as the angle between two radii parallel to the nor- 
mals to the curve at P lt P r If P X P 2 be the subtended arc in the 
standard circle (Fig. 145), the whole curvature of PJP 2 is proportional 
to P X P;, or 




A- $1 = ^-^. 

If the standard circle be taken with unit radius, the entire curvature 
of P X P 2 is measured by the arc s % ' — s x on the unit circle. 

The mean curvature, or average curvature, of P X P 2 is the entire cur- 
vature divided by the length of the curve P x P 2i 

02 - 0i _. V ~ S l 

*t — *i " '. -V 

or, is the quotient of the corresponding arc on the unit circle divided 
by the length of curve P X P 2 . 

The specific curvature of a curve, or the measure of curvature of a 
curve at a point P, is the limit of the mean curvature, as the length 
of the arc converges to zero. It is therefore the derivative of (ft with 
respect to s. But since ds = Rdcp, where R is the radius of cur- 
vature of the curve at a point, we have for the specific curvature 

d<p _ 1 
di ~ ~p' 

The curvature of a curve at a point is therefore properly measured 
by the reciprocal of the radius of curvature. 

To extend this to surfaces, we measure a solid or conical angle by 
describing a sphere with the vertex of a cone as center and radius r. 
Then the measure of the solid angle oo is defined to be the area of the 
surface cut out of the sphere by the cone, divided by the square on 
the radius, or 

A 
«> = &• 

The unit solid angle, called the steradian, is that solid angle which 



37 2 



APPLICATION TO SURFACES. 



[Ch. XXXIII. 



cuts out an area A equal to the square on the radius. In particular, 
if we take as a standard sphere one of unit radius, then 

gj = A, 
or, the area subtended is the measure of the solid angle. 

Definition. — The entire curvature of any given portion of a curved 
surface is measured by the area enclosed on a sphere surface, of unit 
radius, by a cone whose vertex is the center of the sphere and whose 
generating lines are parallel to the normals to the surface at every 
point of the boundary of the given portion of the surface. 

Horograph. — The curve traced on the surface of a sphere of unit 
radius by a line through the center moving so as to be always parallel 
to a normal to a surface at the boundary of a given portion of the sur- 
face is called the horograph of the given portion of the surface. 

Mean or average curvature of any surface. The mean or average 
curvature of any portion of a surface is the entire curvature (area of the 
horograph), divided by the area of the given portion of the surface. 
If £ be the area of the given portion and od the entire curvature, the 
mean curvature is ^ 

Specific Curvature of a surface, or curvature of a surface at a point. 
The specific curvature of a surface at any point, or, as we briefly say, 
the curvature of a surface at a point on the surface, is the limit of the 
average curvature of a portion of the surface containing the point, as 
the area of that portion converges to o. In symbols, the curvature at 
a point is doo 

dS' 

Gauss* Theorem. The curvature of a surface at any point is equal 
to the reciprocal of the product of the principal radii of curvature of 
the surface at the point, or 




*A 



146. 



Let 6" be any portion of a sur- 
face containing a point P. 

Draw the principal normal sec- 
tions PM — As 2 , PN = As v 

.-.js=/is 1 -js. z =p f 1 /i(p 1 -p:^(p 2 

Ago = A<r x > Acr 2 = A(p x >A(p 2 . 
A<j x , Aa 2 being the arcs of the 
horograph corresponding to As x , 
As 2 , on the surface. 

Ago 1 

•'" AS 
In the limit 

dco 1 

dS^PfF,' 



r;K 



Art. 256.] CURVATURE OF SURFACES. 373 



EXERCISES. 

1. Find the principal radii of curvature at the origin for the surface 

2z = 6x 2 - $xy - 6> 2 . Am. f s , - ^. 

2. A surface is formed by the revolution of a parabola about its directrix; show 
that the principal radii of curvature at any point are in. the constant ratio 1 : 2. 

3. Find the principal radii of curvature, at x, y, z, of the surface 

z . z A x 2 4- y 2 4- a % 

y cos x sin — = o. Ans. ± — — — . 

a a a 



4. Show that at all points on the curve in which the planes z = ± — cut 

2ab 

the hyperbolic paraboloid 22 = ax 2 — by 2 the radii of principal curvature of the 
latter surface are equal and opposite. This curve is also the locus of points at 
which the right-line generators are at right angles. 

5. Show from (6), § 248, that the mean curvature of all the normal sections of 
a surface at a point is 



2 U,+ Rj' 



6. Show that at every point on the revolute generated by a catenary revolving 
about its axis, the principal radii of curvature are equal and opposite. 

7. Show that at every point on a sphere the specific curvature is constant 
and positive. 

8. Show that at every point of a plane the specific curvature is constant and o. 

9. Show that at every point on the revolute generated by the tractrix revolving 
about its asymptote, the specific curvature is constant and negative. This surface 
is called the pseudo-sphere. 

10. If the plane curve given by the equations 

x/a = cos S -f log tan -£■ Q, y/a = sin 0, 
revolves about Ox, the surface generated has its specific curvature constant. 

11. If R v R 2 are the principal radii of curvature at any point of the ellipsoid 
on the line of intersection with a given concentric sphere, prove that 

*T+V const . 

12. Prove that the specific curvature at any point of the elliptic or hyperbolic 
paraboloid y 2 /b -\- z 2 /c = x varies as {p/z^, p being the perpendicular from the 
origin on the tangent plane. 

13. In the helicoid/ .= x tan (z/a) show that the principal radii of curvature, 
at every point at the intersection of the helicoid with a coaxial cylinder, are con- 
stant and equal in magnitude, opposite in sign. 

14. Prove that the specific curvature at every point of the elliptic paraboloid 
2z = x 2 /a -\- y 2 /b, where it is cut by the cylinder x 2 /a 2 -j- y 2 /b 2 = 1, is (^ab)-i. 

15. Prove that the principal curvatures are equal and opposite in the surface 
x 2 (y — z) -\- ayz = o where it is met by the cone (x 2 + dyz)yz — (y — z) 4 -. 

16. The principal radii of curvature at the points of the surface 

a 2 x 2 = z 2 K x 2 + y 2 ), ■ where x = y = z, 
are given by 2R 2 -\-'2 4/3 aR — ga 2 = o. 



374 APPLICATION TO SURFACES. [Ch. XXXIII. 

17. Prove that the radius ot curvature of the surface x™ -\- y m -j- z™ = a m at 



m — 2 



an umbilic is 3 2m aj{m — 1). 

x y z 

18. Show that — = — = — is an umbilic on the surface 

a b c 

jfi/a + y 3 /b -f z z /c = k 2 . 

19. Show that x = y = z = (a^)^ is an umbilic on the surface .xyz = a&r and 
the curvature there is \{abc)~^. 

x 2 y 2 z 2 

20. Find the umbilici on the ellipsoid — -\- —4- — = 1. 

a* 0* c 1 

a 2 (a 2 — b 2 ) c 2 (b 2 c 2 ) 

Ans. The four real umbilics are x 2 = —- , z 2 = —^ . 

a 2 — c- a 2 — c' 1 

21. At an ordinary point on a surface the locus of the centers of curvature of all 
plane sections is a fixed surface, whose equation referred to the tangent plane as 
2-plane and the principal planes as the x- and ^-planes, is 



(^ +y* + z 2 ) ^L + >l^ = 2{x 2 + yi) . 



22. Show that an umbilicus on the surface 

(x/a)i + (y /6)* + (z /cf = I 

23. If F = o is the equation of a conicoid, show that the tangent cone to the 
surface drawn from the vertex or, ft, y touches a surface along a plane curve which 
is the intersection of F = o and the plane 

<* " a> ^ + ( -" " /S) ^ + (2 -r)^+1* A x) = o. 

24. Find the quadratic equation for determining the principal radii of curva- 
ture at any point of the surface 

*(■*) + H¥\ + *(*) = o, 

and find the condition that the principal curvatures may be equal and opposite. 

25. Show that the cylinder 

(a 2 4. c 2 )b 2 x 2 + (<5 2 + c 2 )a 2 y 2 = (a 2 + b 2 )a 2 b 2 
cuts the hyperboloid x 2 /a 2 -j- y 2 /b 2 — z 2 /c 2 — 1 in a curve at each point of which 
the principal curvatures of the hyperboloid are equal and opposite. 

26. Show that the principal radii of curvature are equal and opposite at every 
point in which the plane x = a cuts the surface 

x{x 2 + y> + z 2 ) = 2a{x 2 + y 2 ). 

27. In the surface in Ex. 24 show that the point which satisfies 

4>»{x) = rP"(y) = X "(*) 
is an umbilic. 

28. Find the umbilici on the surface 2z = x 2 /a -\- y 2 /b. 



Ans. x = o, y = — \/{ab — b 2 ), z = ±(a — b), if a > b. 
29. Show that z — f(x, y) is generated by a straight line if at all points 

a 2 / a 2 / 

8x 2 ~dy 



f — l— '-A 

V 2 ~ \dx by) 



This is also the condition that the inflexional tangents at each point of the sur- 
face shall be coincident. Such a surface is called a torse or developable surface. 



CHAPTER XXXIV. 
CURVES IN SPACE. 

257. General Equations. — A curve in space is generally denned as 
the intersection of two surfaces. A curve will in general have for its 
equations 

<t> x {x>y, z ) = °, 0,( x >y> z ) = °- C 1 ) 

If between these two equations we eliminate successively x,y, z, we 
obtain the projecting cylinders of the curve on the coordinate planes, 
respectively, 

$i(y> z ) = °> &(■*> z ) = °> t\( x >y) = o. 

Any two of these can be taken as the equations of the curve. 

258. A curve in space is also determined when the coordinates of 
any point on the curve are given as functions of some fourth variable, 
such as /, 

x= 0(/), y = f{t), *=*(/). (2) 

The elimination of / between these equations two and two give the 
projecting cylinders of the curve. 

259. Equations of the Tangent to a Curve at a Point. — If the 

equations of the line are (1), the equations of the tangent line to (1) at 
x,y, z are the equations to the tangent planes to cf) l = o, 2 = o, 
taken simultaneously, or 

Since the tangent line is perpendicular to the normals to these 
planes, the direction cosines /, m, n of the tangent line are given by 

/ m n 1 

vF^raqsTx ~ N ^ - N A ~ L > M 2 - l,m= h - 

where 

** = (Mjr t - M^f + (A\£, - nay + (A*; - L w- 

L,, M v N x being the first partial derivatives of X at x,y, z, and 
similarly Z 2 , M %i N 2 are those of 2 . 

375 



(2) 



(3) 

(4) 



376 APPLICATION TO SURFACES. [Gil XXXIV. 

260. If s is the length of a curve measured from a fixed point 
tox,y, z, then the direction cosines of the tangent to the curve at 
x, y, z are 

dx dy dz 

/= *> '"=Js< *=&' 
and the equations of the tangent are 

X -x _ Y-y Z-z 

dx dy dz 

ds ds ds 
If the equations to the curve be given by (2), § 258, then 

— = (p'(t) — , etc., and the equations (2) become 

X — x _Y — y _Z — z 

In general the equations to the tangent are 
X-x _ Y-y _ Z-z 
dx dy dz ' 

without specifying the independent variable. 

261. The Equation to the Normal Plane to a Curve at x,y, z is 
. __ .dx t/T - . dy dz . , 

<*-*> *+<f -■>>*+ i z -»>.s=°' (i) 

the normal plane being defined as the plane which is perpendicular 
to the tangent at the point of contact. 

Regardless of the independent variable, (1) becomes 

(X - x)dx + (Y-y)dy + (Z - z)dz = o. (2) 

EXAMPLES. 

1. Find the tangent line to the central plane section of an ellipsoid. 
The equations of the curve are 

The equations of the tangent at x, y, z are 

X-x Y — y Z - z 

b 1 c l c l a z a 2 t> 2 

2. Trace the curve (the helix) 

x = a cos /, y = a sin /, z = bt. 

Show that the tangent makes a constant angle with the x, y plane, and that the 
curve is a line drawn on a circular cylinder of revolution cutting all the elements 
at a constant angle. 



Art. 262.] 



CURVES IN SPACE. 



377 



3. Find the highest and lowest points on the curve of intersection of the 
surfaces 

2z = a x 2 -f by 2 , Ax -f By -f- Cz + D = o, 
from the fact that at these points the tangent to the curve must be horizontal. 

4. Show that at every point of a line of steepest slope on any surface F = o 

we must have 

dF , OF J 
■—-. dy — — dx — o. 
dx dy 

5. Show that the lines of steepest slope on the right conoid x = y/(z) are cut 
out by the cylinders x 2 -\- y 2 = r 2 , r being an arbitrary radius. 

262. Osculating Plane. — UP, Q, R be three points on a curve, 
these three points determine a plane. The limiting position of this 
plane when P, Q, R converge to one point as a limit is called the 
osculating plane of the curve at that point. 

The coordinates x,y, z of any point on a curve are functions of the 
length, s, of the curve measured from some fixed point to x, y, z. 
Therefore, if s x be the length to x lt y lt % lf 



dx 



<Px 



!«=* + ('!- O^ + K'l-^-S+W'l - S ) 



ds 



ds 2 



d*x 
do* 9 



where a is the length to some point between x,y, z and x l ,y l , z v 



Put 6s 



D^pc, x" = D\x, etc., then 



x x —_x-\- ds-x' + ±ds*.x" + ISsS.x'J'. 

Let P, Q, R be x, y, z ; x v y lf z 1 ; x 2 ,y 2 , z r Then 
x x = x -f ds-x',,, y\ =y + os-y'^, z l = z + 6s-z' 9 . 
x 2 = x -f- ^^-j; 7 -f- ^tfj 2 • .*£', 



Z == Z -j- /£o>- s' -j- l/£ 2 oV 



(1) 



The equation to the plane through P can be written 

A(X - x) + B{T -y) + C(Z - 2) = o. 

If this passes through <2 and i?, then 

A(x l -x) + B{y x -y) + Cfo - 2) = o, ) 
A(x 2 -x) + B(y 2 -y) + C(* 2 - 0) = o. [ 

Substitute the values of the coordinates from (1) and (2) in (4) 
Divide by ds, 6s 2 , and let o\r(=)o. 

. \ Ax' + .#/ 4- Cz' — o, 
,4.*" + By" -f C*" = o. 



(3) 

(4) 

4). 

(5) 



Eliminating -4, i?, C between (3) and (5), we have the equation to 
the osculating plane at x,y, z, 

(6) 



X- x, 
x' 
x" 



r-y, 

y 

y 



Z -z 
z' 
z" I 



378 



APPLICATION TO SURFACES. 



[Ch. XXXIV. 



Or, regardless of the independent variable, 

X — x, Y — y, Z — z — o. (7) 

dx dy dz 

d 2 x d 2 y dh 

263. To Find the Condition that a Curve may be a Plane 
Curve. — If a curve is a plane curve, the coordinates of any point must 
satisfy a linear relation 

Ax -f By + Cz -f D = o, 

where A, B, C, D are constants. Differentiating, 

Adx -\-Bdy + Cdz = o, 

Ad 2 x -f BcPy + Cdh = o, 

Ad*x + j&/"> + Cdh = o. 

Eliminating ^4,-5, C, we have the condition 

dx, dy, dz 
d 2 x, d 2 y, d 2 z 
d 3 x, d 3 y, d 3 z 

which must be satisfied at all points on the curve. 

264. Equations of the Principal Normal. — The principal normal 
to a curve at a point is the intersection of the osculating plane and the 
normal plane at the point. 

Let /, m, n be the direction cosines of the principal normal at 
x,y, z. Then, since this line lies in the normal and osculating planes, 



y z 



z' x' 
+ m \z"x" 



+ n 

lx' -f- my' -(- nz' : 



x' y 

x"y> 



= o, 



These conditions are satisfied by / = x". m 



since 



y z' 



+y 



+*' 



x y 

x»y 



x"y" z" 
x f y' z' 
x"y" z" 



= o. 



or 



Also differentiating x' 2 -\-y' 2 + z' 2 = 1, 

.'. x f x"+y'y" +z'z" = o. 

Therefore the equations of the principal normal are 
X - x _ Y -y __ Z-z 

x" y" z" 

X-x Y-v Z-z 



d'x 



d 2 y 



d 2 z 



(1) 

(2) 



265. The Binormal. — The binomial to a curve at a point is the 
straight line perpendicular to the osculating plane at the point. 



Art. 267.] CURVES IN SPACE. 379 

Its equations are therefore, from (6), § 262, 

A' - x F- v Z -z 

y - z " _ y " z ' ~ Yx" -z" x' ~ x'y" - x"y'' (4) 

Dividing through by ds 3 , the equations can be written without 
specifying the independent variable. 

266. The Circle of Curvature. — The circle of curvature at a given 
point of a space curve is the limiting position of the circle passing 
through three points on the curve when the three points converge to 
the given point. 

Clearly, the circle of curvature lies in the osculating plane and is 
the osculating circle of the curve. 

To find the radius of curvature. Let a, /?, y be the coordinates 
of the center, and p the radius of the circle of curvature at x, y, z. 
Then 

(* _ af + (y - (5f + {z - yf = p*. 

Let x,y, z vary on the circle. Differentiate twice with respect to s. 
Then 

(*- «)*" + (y- PV + (z - y) z" + x'* + y' > + z f * = o. 

But x' 2 4- y' 2 -[- z' 2 = 1. Also, the line through x,y, z and 
a, ft, y is the principal normal, whose direction cosines, by (1), 
§ 264, are 

<*■" 
/ = 



with similar values for m and n. Since 

x — a = /p, v — fi = mp, z — y = tip, 

The center of the circle is a = x — ip, etc. 
267. The direction cosines of the binormal are 
/ = p[y'z" - z'y"\ m = p{z'x" - x'z"\ n = p{x'y" -/*"). 
For, by (4), §265, 

I _ m _ n . 

v > z " _ z ' y " - z ' x " _ x 'z" ~ x'y" -y'x" * ^ 

Also differentiating x* 2 -\-y f 2 + 2' 2 = 1, 

. '. *'*" + J>>" + *V = O. 

The sum of the squares of the denominators in (1) is 
(*' 2 + y 2 + 2 / 2)( ^/ 2 + y/ 2 + z » 2) _(*'.*" +>/' + s'*") = l/p 2 . 
Hence the results stated. 



380 APPLICATION TO SURFACES. [Ch. XXXIV. 

268. Tortuosity. Measure of Twist. 

Definition. — The measure of torsion or twist of a space curve is the 
rate per unit length of curve at which the osculating plane turns 
around the tangent to the curve, as the point of contact moves along 
the curve. 

If the osculating plane turns through the angle Jr as the point of 
contact P moves to Q through the are As, the measure of torsion at P is 

dr_ _ fAr 
its ~ £ As ' 
when As( = )o. The number <r = D T s is sometimes called the radius 
of torsion. 

Let l xi m lf n l ; I %i m 2 , n v be the direction cosines of two planes 
including an angle 6. Then 

sin 2 # = {m x n 2 - n x m 2 f + («£ - /^ 2 ) 2 + (l x m % - w/ 2 ) 2 . 

Let /, m, n be the direction cosines of the osculating plane at P, 
and / -j- Al, m -\- Am, n -\- An those at Q. 

Let At be the angle between these planes. Then 

sin 2 Z/r = (mAn - nAm) 2 + {nAl - /An) 2 + (/Am - mAf) 2 . 

Divide by As 2 and write 



(1) 



Square this last equation and subtract from (1). 

••■ (f)"= (§h ($)'+ G)' 

269. The measure of torsion can be expressed in another form, as 
follows. 

Let /, m, n be the direction cosines of the binomial, and 
L —y'z" — z' y" , etc., as in § 267. Then 

/ m n 

L=M=N = p - (I) 

Whence Z 2 + M 2 + W = 1/p 2 . (2) 

Since the binormal is perpendicular to the tangent and principal 
normal, 

l x > _|_ m y 4. nz > = o, (3) 

fa" -\- my" -f nz" — o. (4) 







sin 2 At 




sin 2 Ax ( 


JtN 


| 2 












As 2 




Ar 2 \ 


As j 


1 * 








Let' 


As( = ) 


Then, in the limit, 












')*= 


/ dn 

\ m Ts- 


dm\* 
**/ + 


/ dl t 

\ n Ts~ l - 


ds) 


H 


dm 
~ds~- 


— m 


diy 

ds) ' 


Since 


P + m 2 


+ n*= 1, 




■■■ 4 

ds 


4- 


dm 

ds 


dn 

+ n d7-~ 


- 0. 



Art. 270.] CURVES IN SPACE. 381 

Differentiating (3) and using (4), 

/v + »y + »v = oi (5) 

Differentiating, P -f- m 2 -j- «- = 1. (6) 

. '. //' -f mm! -\- ?iri = o. (7) 

From (5) and (7) we get 

/' m' n' 



mz' — ny' "~ nx' — lz' ~~ I'y — mx' 9 
and each of these is equal to 

I'x" m'y" n'z" 



(8) 



mz'x — ny x nx'y" — Iz'y" ly'z" 



' v." — nv'/x" n.Tc'v" — /c'v" ' h>'v" mx'z" 



(9) 



_ I'x" -f m'y" -f n'z" 
lL-\- mM + nN ' 
Differentiating (4), 

l' x " _|_ m 'y" _j_ n'z" -f- &"' -f my'" -f »*'" - . 
Therefore (8) is equal to 

l x "> -j. my'" + nz"' _ x'"L+y'"M+z'"N 
IL + mM+nN ~ L 2 -j- M 2 -f N 2 * 

Remembering that /, m, n; x',y', z' are the direction cosines of two 
lines at right angles, 

{mz' — ny') 2 -j- (nx' — lz') 2 -j- {ly' — mx') 2 
Therefore, by (2), §268, and (8), 

( dT V- ( x'"L+y"M+z'"N \ 2 
\ds) ~ \ L 2 + M 2 -f- N 2 ) ' 



or 



1 dt 

- — — = p 2 

<j as 



x y' z 
x" y" z' 



(10) 



by (2), and the determinant form of 

x'"L -\-y'"M + z'"N. 

270. Spherical Curvature. — Through any four points on a space 
curve can be passed one determinate sphere. The limit to which 
converges this sphere when the four points converge to one as a limit 
is called the osculating sphere, or sphere of curvature. 

Differentiating the equation of the sphere, 

.-. (x-a)x' +(y-/3\/ +(z-y)z' = o, 
(x - a)x" + (y - ft)y" + (z - y)z" = - 1, 
[x - a)x'" + (_>>- V)y'" + (z- y)z'" = o. 



3*2 



APPLICATION TO SURFACES. 



[Ch. XXXIV. 



Eliminating between the last three equations, 



(* 



\y 



z 



= <rp 2 (y'z f// — z'y"); 



x y z 

x" y" z' 

, x'" y" z' 

y — ft - <rp\z'x f " - z f "x')', z - y = ap 2 (x'y'" - x'"y'). 
Squaring and adding, 
R 2 = cr 2 /) 4 [(•*>'" - y'x"'f -f (/*'" - z'y"') 2 + (z'x" f - x'z'"f\. 
Clearly the circle of curvature lies on the sphere of curvature. 
Let P, Q, R ', J be four points on a curve and in the same neighbor- 
hood, R and p the radii of spherical and circular curvature. 

Then, C being the center of the circle through P, Q, R, and £ 

S that of the sphere through 
P, Q, R, J, we have directly 
from the figure, 

Ap 
Ax 



SO 



R 2 



(4ti 



W£)' 



R 2 



'+£)' 





\ 


v v v 


"^^^^i 


ti\ 




\ J 


J c 



r2 



= P' + 



dx 






Fig. 147. 
271. The expressions for the value of the radius of curvature and 
measure of torsion in § 266, and (10), §269, have been worked out 
with respect to s, the curve length, as the independent variable. These 
can be written in differentials, regardless of whatever variable be 
taken as the independent variable. 
Represent by 

dx dv dz 2 
\<Px d 2 y <Pz 
the sum of the squares of the three determinants 

(dycPz - dzd 2 yf -f (dzd 2 x - dx d 2 z) 2 + (dx d^y — dy d 2 x) z . 
Then, regardless of the independent variable employed, 
(dx 2 -f dy 2 -f dz 2 )* 

I II d x dy dz ' 
\| J d*x d*y d*z 



(1) 



dx dy dz 
d?x d 2 y d?z 

(Px d*v d*z 



dx dy dz 
d 2 x d*y d*z 1, 



M 



Art. 27 i.J CURVES IN SPACE. 3. S3 

(1) comes immediately from § 267, and (2) from putting the value 
of p* from (1), § 271 in (10), § 269. 



EXERCISES. 

1. Show that in a plane curve the torsion is o. 

2. The equations of the tangent at x, y, z to the curve whose equations are 



ax* + by* -f cz l = 1, bx 1 + cy 2 -f a? 2 



are 



x(X-x) y(Y-y) z(Z - z) 



ab — c 1 be — a 2 ac — b 2 

3. The equations of a line are 

x 2 _|_ y _j_ s 2 _ 4a 2 an( j ^2 _j_ z 2 _ 2flLr# 

Show that the equations of the tangent line and normal plane are 

{x — a)X+zZ = ax, ) X Y _l a \ ( Z Y \ 

yY+aX = a( 4 a - x) .] '' T ~ ~y~ ~ \* ~ x) \7 ~ ~JJ ' 

4. The equation of the normal plane to the intersection of 

x 2 /a -\- y 2 /b -f z 2 /c = I and x 2 -j- y 2 -\- z 2 = d 2 

X Y Z 

is — a(b — c) -\ b(c — a) -\ da — b) = o. 

x y z K 

5. Show that the curve z(x -f- z)(x — a) = a 3 , z( y -f- z)( y — a) — a 3 , is a 
plane curve. 

6. If the osculating plane at every point of a curve pass through a fixed point, 
prove that the curve will be plane. 

7. Prove that the surface x* -f- y* -f- 2 4 = ^a* cuts the sphere 

x 2 -f y 2 -f z 2 = a 2 
in great circles. 

8. Show that the equations of the tangent to the curve 

y 2 = ax — x 2 , z 2 = a 2 — ax, 

are X - x = 2/ ( Y - v) = - — (Z — 2). 

9. Find the osculating plane at any point of the curve 

x = a cos t, y = b sin /, z = <*A 

Ans. c(Xy - Kr) + ab(Z — z) = O. 

10. Find the radius of circular curvature at any point of 

x/h + y/£ = I. x 2 + * 2 = « 2 . 

(# a 2 -J- £2 S 2)? 



a 2 h 2 j^/h 2 + k 2 

11. Show that the curves of greatest slope to xOy on the surfaces xyz = a 3 and 
£z = xj/ are the lines in which these surfaces are cut by the cylinder x 2 — y 2 = const. 

12. Find the osculating plane at any point of the curve 

x = a cos Q -\- b sin 6, y = a sin -\- b cos Q, z — c sin 20. 



3 8 4 APPLICATION TO SURFACES. [Ch. XXXIV, 

13. Find the principal normal at any point of 

x 2 -f y 2 = a 2 , az — x 1 - y 2 . 
Hint. Express x, y in terms of z as the independent variable. 

14. Given the helix x — a cos 0, y — a sin G, z — bQ, show that 

(i). The tangent makes a constant angle with the xy plane. 

(2). Find the normal and osculating planes, principal normal. 

(3). Locus of principal normals. 

(4). Coordinates of center and radius of curvature. 

(5). Radius of torsion. 

Am. (2). aX sin — a Y cos 6 — b(z — bB) = O, 

bXsin 6 -bVcos S + a(z-bQ) = o. 

V z 

(3).£ = tan-. 

(4). p = «(I + ^V* 2 )- 
(5). o- = (** + **)/*. 

15. Show that F = o, F' z — o are the equations of the line of contact of the 
vertical enveloping cylinder of F = o, and that the horizontal projection of this 
line is the envelope of the horizontal projections of parallel plane sections of F= o. 

16. Show that the equations of the level lines and lines of steepest slope on the 
surface F = o are 

F = o, F' x dx -\- Fy dy — o and F — o, F' x dy — Fy dx — o 
respectively, and that they cross each other at right angles. 

17. Find the lines of steepest slope on the surfaces 

ax 2 -f- by 2 -j- cz 2 = I and s = tf.r 2 -}- £/ 2 . 

18. A line of constant slope on a surface is called a Loxodrone. Find the loxo- 
drone on the cone x 2 -f- y 2 = k(z — c/ 2 . Show that its horizontal projection is a 
logarithmic spiral. 

19. Find the loxodrone on the sphere x 2 -\- y 2 -\- z 2 — a 2 . 

20. A line of curvature on a surface is a line at every point of which the tangent 
to the line lies in a principal plane of the surface. Show that through every 
ordinary point on a surface pass two lines of curvature at right angles. 

21. A geodesic line on a surface is a line whose osculating plane at any point 
contains the normal to the surface at that point. Use Meunier's Theorem to show 
that between two arbitrarily near points on the surface the geodesic is the shortest 
line that can be drawn on the surface. Show that at every point on a geodesic on 
the surface F = o, we have 

F' x /d 2 x = F' y /d 2 y = Fi/dH. 



CHAPTER XXXV. 

ENVELOPES OF SURFACES. 

272. Envelope of a Surface - Family having One Variable 
Parameter. — When F(x,y, z) = o is the equation of a surface con- 
taining an arbitrary parameter a, we can indicate the presence of this 
arbitrary parameter a by writing the equation 

F(x,y, z,a) = o. (1) 

The position of the surface (1) depends on the value assigned to a. 
By assigning a continuous series of values to awe have a singly infinite 
family of surfaces whose equation is (1). 

If we assign to a a, particular value a v we have another position of 
the surface (1) whose equation is 

F(x, y, z, a x ) = o. (2) 

The two surfaces (1) and (2) will in general intersect in a curve. 
When a^=.)a the surface (2) converges to coincidence with the sur- 
face (1), and their line of intersection may converge to a definite posi- 
tion on (1). At any point on the intersection of (1) and (2) the 
values of x,y, z are the same in both equations. By the law of the 
mean, 

F(x,y, z, a x ) = F(x,y, z, a) + (a x — a)F' a ,(x,y, z, a'), 

a' being a number between a and a v 

At any point of intersection of (1) and (2) 

F(x,y, z, a) = F(x,y, z, a x ) = o. 

Therefore at any such point we have 

F«,(x,y, z, a') = o. (3) 

If, when a x (=)(x, the line of intersection of (1) and (2) converges 
to a definite position on (1), then the coordinates of all points on this 
line must satisfy, by (3), the equation 

— F(x,y, z, a) = o, (4) 

and the surface (4) passes through the limiting position of (1) and (2). 
If from equations (1) and (4), i.e., 

F{x,y, z, a) = o, F' a {x,y, z, a) = o, (5) 

385 



386 APPLICATION TO SURFACES. [Cn. XXXV. 

a be eliminated, the result is an equation cf)(x, y, z) = o, which 
is the surface generated by the line whose equations are (5), or = o 
is the locus of the ultimate intersections of consecutive surfaces of the 
family (1). This locus is called the envelope of the family (1). The 
line whose equations are (5) is called the characteristic of the envelope. 

273. Each Member of a Family of One Parameter is Tangent 
to the Envelope at all Points of the Characteristic. — The parameter 
a being assigned any constant value, the tangent plane to 

F(x,y, z, a) — o, at x,j>, z, is 

dF , dF 7 dF J 

— dx + - Jr dy + ^-dz= o. (1) 

dx dy dz v y 

But in F(x,y, z, a) = o, as x,y, z vary along the envelope, a also 
varies, and the equation to the tangent to the envelope is 

dF 7 dF J dF ' dF 7 

— - dx -f- — - dy -f — - dz -f —- da — o. (2) 

dx dy dz da v ' 

Since at any point x,y, z common to the surface F= o and the 
envelope, that is all along the characteristic, we have F* = o, the 
planes (1) and (2) coincide. 



EXAMPLES. 

1. Show that the envelope of a family of planes having a single parameter is a 
torse (developable surface). 

Let z = x(P(a) +)'ip(a) -f *(<*). 

dz dz lit 

.-. — = 4>(a), -- = tp(a); :• x<p a + yif> a + Xa = °- 

Also, 

&z . t da &z da dH da ... da 

w=--^ a) ^' dy = 1j){a) dj' -d7dj =<p{a) ^y =i,{a) a*- 



Hence rt — s' 2 = o. See Ex. 29, § 256. 

Ans. Hyperbolic cylinder, xz -\- yz = \. 



« t- 1 x 4- v , 

2. Envelop — — \- za = 2 

a 



3. Envelop x -{- y — iaz = a 2 . 

Ans. Parabolic cylinder, x -j- y -f- - 2 = o. 

4. Generally if <p, ip, x are linear functions of x, y, z, then the envelope of the 
plane 

<pa 2 + 2ipa + x = ° 
is ip' 2 = <px, a cone or cylinder having <p = o, j - o as tangent planes, and 
if) — o is a plane through the lines of contact. 

5. Find the envelope of the family of spheres whose centers lie on the parabola 
x- -f 4ay = 0, z = o, and which pass through the origin. 

Ans. x 2 -f- y 2 -f- z 2 = 2ax' : y. 



Art. 274.] ENVELOPES OF SURFACES. 387 

6. Find the envelope of a plane which forms with the coordinate planes a 
tetrahedron of constant volume. 

Ans. xyz = const. 

7. Find the envelope of a plane such that the sum of the squares of its 
intercepts on the axes is constant. 

Ans. x^ -\- y% -\- zl — const. 

274. Envelope of a Surface-Family with Two Variable Param- 
eters. 

If F(a i p) = F(x t y i z,a 3 p) = o (1) 

is a surface of the family, then 

F(a lf fi,) = F(x,y,z, a l9 fi,) = o (2) 

is a second surface of the family. 

At any point x, y, z where (1) and (2) meet, 

F(a v X) = *\a, fi) + (a, - «)g+ {fi, - /J)|j£, (3) 

where a' is between a l and a, fi' between (3 X and fi. 
In virtue of (i) and (2), (3) gives 

(«.-«>!£+ (ft- *>?£ = «»• (4) 

This is the equation of a surface passing through the intersection 
of (1) and (2). But for any fixed values x,y, z, a, fi satisfying (1) 
and (2) there are an indefinite number of surfaces (4) obtained by 
varying a lt fi x , all of which cut (1) in lines passing through x, y, z. 
Consequently there are of these surfaces (4) two particular surfaces, 

dF _ dF _ 

da'~~ °' dfi 7 ~~ °' 

which cut (1) in lines passing through x, y, z. 

If now the point x,y, z has a determinate limit when a x (-=)a, 
fi x ( = )fi, then the three surfaces 

F(a, fi) = o, F' a (a, fi) = o, F&a, fi) = o, 

pass through and determine that point. 

These surfaces (5) intersect, in general, in a discrete set of points. 
If, however, we eliminate between them a and fi, we obtain the equa- 
tion to the locus of intersections. This locus is a surface called the 
envelope of the family (1). 

275. The Envelope of the Family F(x,y, z, a, fi) = o is Tan- 
gent to Each Member of the Family. — The tangent plane to any 
member of the family is 

dF 7 dF 7 dF , , x 

_■**+ - dy + aT ^ = o. (1) 



3^8 APPLICATION TO SURFACES. [Ch. XXXV. 

As the point x, y, z moves on the envelope, a and /3 vary. The 
plane tangent to the envelope is 

dF dF dF dF dF 

^ x + T/» + aT* + a^ da + ^ dP = °- (2) 

At a point x,y, z common to the envelope and one of the surfaces, 
we have F' a = o, Fp = o, and therefore the planes (i) and (2) 
coincide. Since this point is the intersection of the line whose 
equations are F' a = o, Fp = o with the surface F = o, the envelope 
is tangent to the surface at a point, and not along a line. 

276. Use of Arbitrary Multipliers. — K F(x,y, z, a, /3) = o, 
where a, (i are two arbitrary parameters connected by the relation 
(p(a, /?) = o, then F = o is a family of surfaces depending on a 
single variable parameter. The equation of the envelope is found by 
the elimination of a, ft, da, d/3 between 

F = o, = o, F' a da-\- F'pd/3 — o, cp' a da + tpfrd/3 — o. 

This is best effected, as in the corresponding problems of maximum 
and minimum, by the use of arbitrary multipliers. Thus the equation 
of the envelope is the result obtained by eliminating a, [3, X from 

F= o, = o, F' a + \0' a = o, F'p + A0£ = o. 

The family of surfaces, represented by F = o, containing n 
parameters which are connected by n — 1 or n — 2 equations is 
equivalent to a family containing one or two independent parameters 
respectively. Such a family, in general, has an envelope. The 
problem of finding the envelope is generally best solved by intro- 
ducing arbitrary multipliers to assist the eliminations. 

If more than two independent variable parameters are involved, 
there can be no envelope. Foi in this case we obtain more than 
three equations for determining the limiting position of the intersec- 
tion of one surface with a neighboring surface. From these three 
equations x,y, z could be eliminated, and a relation between the 
parameters obtained, which is contrary to the hypothesis that they 
are independent. 

In general, if F— o contains n arbitrary parameters a x , . . . , a n 
connected by the n — 1 equations of condition X = o, . . . , 
M _j = o, the equation of the envelope is found by eliminating the 
27i — 1 numbers a x , . . . , a n , X lt . . . , A M _,, between the 2ti 
equations 

F = O, 0, = O, . . . , 0„_, = o, 

*"K) + WK) + • • • +A*_i<ft-xfo) - o, 



F\a n ) + A x 0/(^) + . . . + Vt0U(««) - o. 



Art. 276.] ENVELOPES OF SURFACES. 389 



EXERCISES. 

1. Find the envelope of (9 being the variable parameter) 
x sin 9 — y cos 9 = aQ — cz. 



/*;/j. ^ sin y cos ! — " ±-±- = a/x 2 4- v 2 -a 1 

a a r \ s 

2. Find the envelope of a sphere of constant radius whose center lies on a circle 
in the .rv-plane. 

Ans. If x 2 -\-y 2 = c 2 is the circle, and the sphere has radius a, the envelope is 
the torus x 2 4- y* = [c 4- (a« - «»)*]*. 

3. Find the envelope of the ellipsoids -- 4- *— 4- ~ = 1, where a -j- ^ + c = k. 

Ans. x* -\- y%-{- z% = k*. 

4. Find the envelope of the ellipsoids in Ex. 3 when they have a constant 
volume. 

5. Find the envelope of the spheres whose centers are on the jtr-axis and whose 
radii are proportional to the distance of the center from the origin. 

Ans. y 2 + z 2 = m 2 (x 2 -j- y* + z 2 ). 

6. Find the envelope of the plane ax -+- fiy -f- yz = 1 when the rectangle 
under the perpendiculars from the points (a, o, o) and (— a, o, o)on the plane is 
equal to k 2 . 

x 2 y 2 4- z 2 

Ans. a, fi positive. ^-^ + J 2 = I. 

7. Find the envelope of the planes J- ~~ -\ =1 when a H 4- b n -f- c n = k n . 

n n n n 

Ans. x n+I ±y n+I + z^ 1 - k*+ s . 

8. Spheres are described having their centers on — = — = — , and their radii 

I m n 

proportional to the square root of the distance of the centers from the origin ; show 
that the equation of the envelope is (/, m, n being direction cosines) 

x 2 4-jv 2 4- z 2 = {lx 4- my 4- nz 4- c) 2 . c = const. 

9. Envelop the family of spheres having for diameters a series of parallel chords 
of an ellipsoid. 

10. If f{cc) = o is the equation of a family of surfaces containing a single 
arbitrary parameter a, then the equations of the characteristic line on the envelope 
are F^a) = o, F\a) = 0. As a varies this line moves on the envelope; it will in 
general have an enveloping line on that surface. The envelope of the characteristic 
is called the edge of the envelope. Show that the equations of the edge of the enve- 
lope are obtained by eliminating a between 

F{q) = o, F'(a) = o, F"(a) = o. 

11. Find the equations of the edge of the envelope of the plane 

x sin 9 — y cos 9 = aH — cz. 

cz 
Ans. x 2 4- y 2 = a 2 , y — x tan — . 

a 

12. Envelop a series of planes passing through the center of an ellipsoid and 
cutting it in sections of constant area. 



39Q APPLICATION TO SURFACES. [Ch. XXXV. 

Let Ix ~\- my -(- nz = o be the plane; the parameters are connected by 

,2 _|_ m * _j_ „2 — I, A ? 2 _|_ m 2p _^_ w 2^a _ ,/2. 



JC 



^» J - t» — : s + t^—t> + 



fl 2 _ dl ' <fr 2 _ ( f2 ' ,2 ^fp— 

13. Spheres are described on chords of the circle x 2 -\- y 2 = 2ax, z=zO which 
pass through the origin, as diameters, show that they are enveloped by 

(x 2 -f y 2 -{- z 2 — ax) 2 = a 2 (x 2 -f y 2 ). 

14. Show that the envelope of planes cutting off a constant volume from the 
cone ax 2 -J- by 2 -j- cz 2 = o is a hyperboloid of which the cone is the asymptote. 

15 Find the envelope of the plane Ix -f- my -{-nz = d, when I 2 -\- m 2 -\- n- = I, 
I 2 n 



+ ^2 ~2=°- 



d* — a 2 ' d 2 — b 2 ' d 2 - 
Ans. Fresnel's Wave-surface, 

a 2 x 2 b 2 y 2 



X 2 _f_ y% _|_ Z 2 _ a 2 ' x 2 _^ y 2 _|_ 2 2 __ &2 I x 2 _|_ y 2 _j_ 2 2 _ c 1 

16. Find the envelope of a plane passing through the origin, having its direc- 
tion cosines proportional to the coordinates of a point on the line in which intersect 
the sphere and cone 

x i _j_ yi 4- Z '2 - r 2 , x 2 /a 2 -f y 2 /b 2 -f z 2 /c 2 = o. 

17. Find the envelope of a plane which moves in such a manner that the sum of 
the squares of its distances from the corners of a tetrahedron is constant. 

18. Show that the envelope of a plane, the sum of whose distances from n fixed 
points in space is equal to the constant /', is a sphere whose center is the centroid 
of the fixed points and whose radius is one «th of k. 

19. Show that the envelope of a plane, the sum of the squares of whose distances 
from n fixed points in space is constant, is a conicoid. Find the equation of the 
envelope. 

20. If right lines radiating from a point be reflected from a given surface, the 
envelope of the reflected rays is called the caustic by reflexion. 

Show that the caustic by reflexion of the sphere x 2 -f- y 2 -f- z 2 = r 2 , the radiant 
point being h, o, o, is 

[tfi 2 p 2 - r 2 (p 2 + 2hx -f h 2 )Y = 2Th\y 2 + z 2 ){p 2 - /i 2 ) 2 , 
in which ft 2 = x 2 -\- y 2 -J- z 2 . 



PART VII. 

INTEGRATION FOR MORE THAN ONE VARIABLE. 
MULTIPLE INTEGRALS. 

CHAPTER XXXVI. 
DIFFERENTIATION AND INTEGRATION OF INTEGRALS. 

277. Differentiation under the Integral Sign. Indefinite In- 
tegral. — ~Let/(x, y) be a function of two independent variables x, y. 
Let 

F(x,y) = j/(x,y)dx, 

the integration being performed for y constant. This integral is a 
function oty as well as of x. On differentiating with respect to x, 

Again, differentiating this with respect toy, 
d*F = d f(x,y) 
dy dx dy 



But 



d 2 F d 2 F 9/ 



Consequently 



dx dy dydx dy 
dy dy 



dF 
d~V 



=/%«■ 



or 



lf Ax ,yY* = p f ^ dx . 



Therefore, to differentiate with respect to y the integral taken with 
respect to x of a function of two independent variables x, y, differ- 
entiate the function under the integral sign. 

39' 



392 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVI. 
In like manner we have 

This process is useful in finding new integrals, from a known inte- 
gral, of a function containing an arbitrary parameter. 

EXAMPLES. 

1. We have the known integral 

/eaxdx = — . 
a 

Differentiating with respect to a, we find 
And generally, differentiating n times, 

_ c . p . , cos ax 

2. Since I sin ax ax = , 



-I 



x sin ax cos ax 
x cos ax ax = — . 



(a + bx)*dx = — -P- -rr show that 



/, s , (nbx — a)(a 4- 6x)* 

x(a + bxY-^dx = 



x a dx = r - z show that 



n(n + i)b 2 
x<*+* 



J^logx&=^ i (logx--L) 



278. Differentiation of Definite Integrals when the Limits are 
Constants. 

f(x,y)dx, 

where a and b are independent of x. Then the result of § 277 holds 
as before, and 

On account of the importance of this an independent proof is 
added. Let Au denote the change in u due to the change Ay \ny. 
Then, the limits remaining the same, 

<*« =£lf(*>y + 4) -/(-v.^)]</-v. 
4 s . . r /(*>->' + 4) - (A. >0^ 



9« _ r> 



£ & 



Ay J a Ay 



Art. 279.] DIFFERENTIATION OF INTEGRALS. 393 

Hence, when 4>'( = )o, we have 
du 



and, generally, 



du /** df 

~1 i dx ' 



d n u _ r b ay 
dy»~Ja dr 



^ dx . 



EXAMPLES. 

1. K r-e-axdx = I 

Jo a 

be differentiated n times with respect to a, we get 

1 n \ 

x n e~* x dx — — 1 



1: 



no 






or 



2. From r dx = -?- 

«/o U 2 + «) " 2ai 

Z* 00 <**" _ 1-3-5 . . • (2» — i) ?r 

X (* 2 + a ) n+t ~ 2.4.6 ... 2« 2 a w+ ±' 
The value of a definite integral can frequently be found by this method. Thus: 

3. Let u = f* ( f" ~ l ) -dx. 

Jo lo S * 

Then * = f l ^ **£* = Z* 1 W. = _L_. 

«fa J log x J a 4- I 

5^-j = '<>? (« + '). 
constant being added since u = o when a = o. 

4. Find /""log (1 4- a cos 0)^9. 

^/w. ar log (1 + ^1 - a 2 ). 

279. Integration under the Integral Sign. 
I, Indefinite Integral. 
Let F(x,y)=f f(x,y)dx. 

Then will 

f[fA*>s)**] & = f[ff( x > y¥y\ dx - 

Let v = / y(jtr,j^)^j/. 

Then ~ =f(x,y). 

Also, ty$ v dx = / V ^ r = fA x >y) dx = F i x >y)- 

.'. fvdx =JF(x,y)dy y 

f I JA*>yyy \ dx =f J jf(x>y) d * \ d y- 



394 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVI. 

Hence the order in which the integration is performed is indif- 
ferent. This shows that in indefinite integration when we integrate 
a function of two independent variables, first with respect to one vari- 
able and then with respect to the other, the result is the same when 
the order of integration is reversed. This being the case, we can 
represent the result of the two integrations by either of the compact 
symbols 

f $/*<* = f }/***■ 

As in differentiation, the operation is to be performed first with 
respect to the variable whose differential is written nearest the func- 
tion, or integral sign. 

II. The same theorem is true for the definite double integral of a 
function of two independent variables when the limits are constants. 
Let 

ff(x, y)dx = X(x, y), ff(x, y)dy = F(x, y) ; 

fff(x,y)dxdy =fff(x,y)dydx = F(x,y). 
Then 

p/dx = X{ x% ,y) - X{x v y), 

pfdy = T(x, ,.,) - F(x, A ); 

i/yi 

•C 1 !n fdy \ dX = F{X "- h) - r(x >' J > ] - F(X *' y > ] + *l*»-ti' 
The last two values are the same. Hence 

C\r;y dx \ dy =i:;\s:y d A^ 

or 

P pA*, yY* *» = P P A*.y)<iy **■ 

The integral sign with its appropriate limits and the corresponding 
differential are written in the same relative position with respect to 
the function. 

EXAMPLES. 



1. f l x a ~ I dx=~. Hence 

Jo a 

Jo Ja J ao a a o 

*i .*■*,- 1 _ x a. 2 -i a 

I 1 </jf=log-l. 

Jo log x a 



Art. 280.] DIFFERENTIATION OF INTEGRALS. 395 

Put x = c~*. 

/ — dz = log-i. 

Jo a o 

2. I ^ a ' v sin />.r dx = -5 7- . 

Jo °- + b 

C^ P a , r*\ b da 

/ / l *-«* sin 6x da dx = / , 

Jo J„ Ao « 2 + * 2 ' 



J/tco^-,, 




or 
If a =r o, a x = 00 , then 



3. Evaluate 
Put 



— sin ^x <Zr = tan-i -± — tan- 1 -A 



r*> sin /^ 




f-^*- 




k= r e~* % dx. 








r e -aHt+x*) a dx _ k - a \ 




r Fe-«^+**)adadx = kfe-< 

Jo Jo Jo 


*V« = k\ 


fe-^+^ada^ 1 * 2 
Jo 2 1 -j- x l 




Vo i-h^ = ^ tan ^J - i7r = 


&. 


He-^dx = \ \/n, 




Jo 2a 





and 

Also, 
and 

and 

This gives the area of the probability curve. 

280. If F(x, y, z) is a function of three independent variables, the 
same rules as for a function of two independent variables govern the 
triple integral 

j j JFdxdydz. 

Examples of double and triple integrals will be given in the next 
chapter. 



CHAPTER XXXVII. 



APPLICATIONS OF DOUBLE AND TRIPLE INTEGRALS. 
Plane Areas. Double Integration. 

281. Rectangular Formulae. — If x, y are the rectangular coor- 
dinates of a point in a plane xOy, then 

doj = Ax Ay = dx dy 

is an element of area, being the area of the rectangle whose sides are 

Ax and Ay. 

Let the entire plane xOy be 
divided into rectangular spaces 
by parallels to Ox and Oy, of 
which Ax Ay is a type. The 
area of any closed boundary 
drawn in the plane is the limit 
of the sum of all the entire rectan- 
gular elements of type Ay Ax 
included in the boundary, when 
for each rectangle Ax( — )o, 
Ay{ — )o. For the area within 
the closed boundary A is equal to 

A = 2 Ay Ax 
plus the sum of the fractional 
rectangles which are cut by the 
Fig. 148. boundary. This latter sum can 

be shown to be less than the length of the boundary multiplied by the 

diagonal of the greatest elementary rectangle, and therefore has the 

limit zero. Hence 

A =£2 Ay Ax, 

taken throughout the enclosed region, when Ax( = )Ay( = )o. 

The summation is effected by summing first the rectangles in a 
vertical strip PQ and then summing all the vertical strips from R to T; 
or, first sum the elements in a horizontal strip PL, then sum all the 
horizontal strips in the boundary from Sto U. These summations are 
clearly represented by the double integrals 



y 




t 








u 




u 




^ ^d 






s 


N 






/ 

/ 




\ 




7 




\'t 




7 




Y • 




1 








M 




1 




-? 


7 














Xz : 


7 




L\ 


*=^== 


: lllTz 




V 




■e^ 








w 


, X 





r 


M 


i 



r' r* M <fy dx, P" f' =Hf) dx dy. 



396 



Art. 282.J APPLICATIONS OF DOUBLE AND TRIPLE INTEGRALS. 397 



In the first integration in either case the limits of the integration 
are, in general, functions of the other variable which are to be deter- 
mined from the given boundary. 

EXAMPLES. 

1. Required the area between the parabola y 2 = ax and the circle y 2 = 2ax — x 2 , 
in the first quadrant. 

The curves meet at the origin and at x = a. 



(1). 



A = T f y= V _l ax - x \/y dx = T[ \/2ax - x 2 - i/2^] dx 
Jo J y— \'ax Jo 



1 y 
no 2 2a 



(2). 



_ /*?=<* Px=y?-/a 
J y=o Jx=a— Vrt 2 



_dx dy 
x 



7Ca L 2a 1 

~4 3 



2. Find the area outside the parabola y 2 — $a(a — x) and inside the circle 

y 2 = 4a 2 - x 2 . 

3. Find the area common to the parabola 3V 2 = 2$x and 5.x- 2 = 91'. Ans. 5. 

282. Polar Coordinates.— The surface of the plane is divided into 
checks bounded by rays drawn 
from the pole and concentric 
circles drawn with center at the 
pole. 

The exact area of any check 
PQ bounded by arcs with radii p, 
p -f- Ap, and these radii includ- 
ing the angle A 6, is 

i{( P + Apy-p*}je 

= pApA6 + ±Ap*Ad. 

The entire area in any closed 

boundary is the limit of the sum 

of the entire checks in the 

boundary. The sum of the par- 




Fig. 149. 






tial checks on the boundary being o when Ap{ — )A6{ — )o, as in § 281. 
But, since 

/ pAp A8 + \Ap>A6 _ £Ap 

jAiAO - I+ U p' 

= *i 
when Ap( = )A£)( = )o, the area within any closed boundary is equal to 

A = £2pApAd 
when Ap( = )A6(=)o. 



398 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cm XXXVII. 



This summation can be effected in two ways : 

(i). We can sum the checks along a radius vector R S, keeping 
A 6 constant, then sum the tier of checks thus obtained from one value 
of 6 to another. 

(2). We can sum the checks along the ring UV, keeping p and 
Ap constant, then sum the rings from one value of p to another. 

These operations clearly give the double integrals 



«=m(p) 



JfOi /»P=«PW , , a /•Pa /»»=mp; JQ . 

' / pdpdd, / / ddpdp. 

EXAMPLES. 

1. Find the area between the two circles p = a cos 0, p = b cos 0, b > a. 

IT 
/» 2 /»£ COS 
(I). ^ = / / P ^P <#, 

Jo Ja cos 
_ r 2 ^(£2 _ fl 2) COS 20 dQ =j (b 2 - c 2 ). 

-IP -»-£ 

ncos -7 /»« /»COS A 

* dB pdp+ / P <# P *P, 

Jo Jcos -1 ^ 
which gives the same result as (1). 

The double integration is not necessary for finding the areas of 
curves; it is given here as an illustration of a process which admits of 
generalization. 

Volumes of Solids. Double and Triple Integration. 

283. Rectangular Coordinates. — 

Let x,y, z be the coordinates of a 
point in space referred to orthogonal 
coordinate axes. 

Divide space into a system of rectan- 
gular parallelopipeds by planes parallel 
to the coordinate planes. Let Ax, Ay, Az 
be the edges of a typical elementary 
parallelopiped. Then the volume 
x Ax Ay Az 

is the elementary space volume. 

The volume of any closed surface is 
the limit of the sum of the entire elemen- 
tary parallelopipeds included by the 
surface when Ax( = )Ay( = )Az( = )o. 

V = £2 Ax Ay Az, 




taken throughout the enclosed space. 
(1). Let x, y, Ay, Ax be constant. 



Sum the elementary volumes 



Art. 283.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 399 

between the two values of z, obtaining the volume of a column MS of 
the solid. The result expresses z as a function of x and r given by 
the equation or equations of the boundary. 

(2). Let x, Ax be constant. Sum the columns between two 
values of y for Ay( — )o. The result is the slice of the solid on the 
cross-section x = constant, having thickness Ax. 

(3). Sum the slices between two values of x for Ax(=z)o. The 
result is the total sum of the elements, expressed by the integral 

Jxi Jy=<j>(x) Jz — K{x,y) 



f x * r*z dy 

£y x dx. 



dx. 



Clearly, if more convenient we may change the order of integra- 
tion, making the proper changes in the limits ot integration. 

EXAMPLES. 



1. Find the volume of one eighth the ellipsoid 

^ + ^- + --1. 






Jo Jo 



x y j j 



n II \~ ^/ dx = * 7tabc ' 



\ 

be 
4 

2. Find the volume bounded by the hyperbolic paraboloid xy = az, the xOy 
plane, and the four planes x = x x , x = x 2 , y = y x , y = y 2 . 

xy 

V= [**[** fdzdydx, 

Jx 1 Jy x Jo 



Jx x Jy x a 



J Xl 2a 



2 
x dx, 



I 

\a 

= — (x 2 — x x )(y 2 - v l )(x 1 y l + x 2 y 2 -f x x y 2 -f x 2 y x \ 
t\a 

= i{x 2 - X X ){X 2 -y x \Z x + 2 2 +^ 3 + **)• 



4oo INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVII. 




The volume is therefore equal to the area of the rectangular base multiplied by 
the average of the elevations of the corners. This is the engineer's r.ule for calcu- 
lating earthwork volumes. 

Polar Coordinates. — The polar coordinates of a point P in 
space are p, the distance of the point from 
the origin; 0, the angle which this radius 
vector OP makes with the vertical Oz; and 0, 
the angle which the vertical plane POz makes 
with the fixed plane xOz. 

Through P draw a vertical circle PM 

with radius p. Prolong OP to R, PR = A p. 

■x Draw the circle R Q in the plane POM with 

radius p -f dp. If A A is the area PRQS, 

then 

/ AA _ 
AppA6~ l ' 

We may therefore take dA = p dp dd. This area revolving around 
Oz generates a ring of volume 

2 7T p sin 6 dA. 

Therefore the volume generated by dA revolving through the arc 
ds — p sin 6 dcp is in the same proportion to the volume of the ring as 
is the arc to the whole circumference, or the element of volume is 

p 2 sin 6 d(p dp dd. 

We divide space into elementary volumes by a series of concen- 
tric spheres having the origin as center, and a series of cones of 
revolution having Oz for axis, and a series of planes through Oz 

The volume of any closed surface is the limit of the sum of the 
entire elementary solids included in the surface when 

Ap{ = )A<t>{ = )A0( = )o. 

Or, the volume is equal to the triple integral 

V = f ffp 2 sin 6 d<f> dp dd, 

taken with the proper limits as determined by the boundaries of the 
surface. 

EXAMPLES. 



1. Find the volume of one eighth the sphere p = a 

n it 



V- f 2 f 2 ( P a p 1 dp • sin QJO.Jfa 

= - 2 / * smBdB'd<t> 

3 J<t>=o Je=o 

IT 



Art. 285.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 401 

The first integration gives a pyramid with vertex O and spherical base 
a 2 sin i/0 </(p. The next integration gives the volume of a wedge-shaped element 
of a solid between two vertical planes determined by <p and <p -\- A (p. The last 
integration sums up these wedges. 

2. A right cone has its vertex on the surface of a sphere, and its axis coincident 
with the diameter of the sphere passing through the vertex; find the volume 
common to the sphere and cone. 

Let a be the radius of the sphere, a the semi-vertical angle of the cone. The 
polar equation of the sphere with the vertex of the cone as origin is p = 2a cos G. 

/»2ir /»a /*2/i cos B 

.-. V=\ I / p 2 dp-sint dd-d<p. 

Jo Jo Jo 

3. The curve p = a(l -(- cos 0) revolves about the initial line; find the volume 
of the solid generated. 

/*«• / , 2tt /»a(i -1- cos 9) 

V— I I I p 2 dp. dip. sind d9, 



pit p21T /*< 

Jo Jo Jo 

/ (I + C0S Q f Sin & dO = §7T<Z 3 . 



27ta* 



285. Mixed Coordinates. — Instead 
of dividing a solid into columns stand- 
ing on a rectangular elementary basis, 
as in the method 

V — f (zdxdy, 

it is sometimes advantageous to divide 
it into columns standing on the polar 
element of area. Thus the elementary 
column volume is 

z p dd dp. 
Therefore for the volume of a solid 
we have 

V= fffdz-pdddp, 

= I I z pdpdti, 
taken between the proper limits. 




Fig. 152. 



EXAMPLES. 

1. Find the volume bounded by the surfaces z = o, 

x2 ~h y 2 = 4- az an< ^ y 1 — 2CX ~ x2. 

Here z = p 2 J\a and the limits of p and must be such as to extend the inte- 
gration over the whole area of the circle y 2 = 2cx — x 2 . Let p x = 2c cos 0; 
then 

J-frr Jo A a a J -in 

it _ ^itc* 
a 4 2 2 — Sa 



V 



= — / cos 4 S df) = — 
a J 



40 2 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVII. 



2. Find the volume of the solid bounded by the plane z = o and the surface 



z = ae c* . 



— xij^y-i, ... V=afft c<i pdpdB. 
f^e '* pdp = - \<*e '*J° = \cK 

/ dB - 2rt. 
Jo 



V = nac\ 



[See Todhunter, Int. Cal. p. 181.] 



Surfaces of Solids. 

286. When the plane through any three points on a surface (the 
points arbitrarily chosen) converges to a tangent plane as a limit when 
the three points converge to a fixed point as a limit, then a definite 
idea of the area of the surface can be had, as follows : 

Inscribe in a given bounded portion of the surface a polyhedral 
surface with triangular plane faces. The area of the given portion of 
the surface is the limit to which converges the area of the polyhedral 
surface when the area of each triangular face converges to zero. 

To evaluate the limit of the sum of the triangular areas inscribed in 
the surface we proceed as follows: 

Let P be a point x,y, z on a surface, 
and Q a point x -f- Ax,y -\- Ay, z -j- Az. 
The prism MTNU on the rectangle 
whose sides are Ax, Ay cuts the surface 
in an element of surface PRQS. Draw 
the diagonal JAY and the two inscribed 
triangles PRQ and PSQ. Let perpendic- 
ulars to the planes of the triangles PRQ, 
PSQ at the point jPmake angles y v y % 
with Oz respectively. The angles y x , y 
are then the angles which these planes make with the horizontal plane 
xQy. Since the area of the orthogonal projection of a plane triangle is 
equal to the area of the triangle into the cosine of the angle between 
the plane of the triangle and the plane of projection, we have the areas 

PRQ = J/TXsec y x , PSQ = J/^lVsec y t . 

Also, MTN=MUN. 




PRQ + PSQ = ^ C ^+ SeCr 



Ay Ax. 






Art. 286.J APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 403 



By hypothesis, if A 2 S is the area of the element of surface PRQS. 



l.ien 



/ 



A*S 



sec y x + sec y 7 



Ay Ax 



But when Q( — )P the perpendiculars to the planes PRQ, PSQ 
have the normal to the surface at P as a limit, since the planes PRQ, 
PSQ converge to the tangent plane at P as a limit. 

If y is the angle which the tangent plane at P makes with the 
plane xOy, then 

sec y -f sec v 

— — sec y, 



I 



-<• +(£)'+ (©■ 



</ 2 S 



and 



= / / sec y <#/ </#, 



sec y, 



=i/V 



. + ffiY+fflV* 



W 



taken between the limits determined by the boundary of the portion 
of the surface whose area is required. 






EXAMPLES. 

1. Find the area of the sphere-surface x 2 -f y 2 + z 2 = a 2 . 
dz x dz- y 

dx z ' by ~ z 

* y = tfa?-* dy dx 



J x=o J y- 



Tta /»« 

= — / dx — \na 2 . 
2 Jo 



2. The center of a sphere whose radius is a is 
on the surface of a cylinder of revolution whose 
radius is £ a. Find the surface of the cylinder 
intercepted by the sphere. 

(1). Let the equations of the sphere and 
cylinder be 



y 2 + z 2 = a\ 



x 2 -j- y 2 



as in the figure. 




4©4 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIL. 



5 = 



l/rt 2 -<«A; f/2 dx 



y ax — x l 



f a x/d^rVx, C a la . 

i I ~1 _ — dx = 2a I ^\—dx = 4a*. 

Jo \/ax-x 2 Jo \ x 



yax — x- 

(2). Let s be the length of the arc of the base of the cylinder measured from 
the origin. Then 

S = 4 fz ds t 

taken over the semi-circumference. Let y be the angle which the sphere radius to 
P makes with Oz, and 6 the angle which OM = p makes with Ox. Then 

z = a cos y = a sin Q. s = aQ, ds = a dQ. 

. •. .9 = 4/ a 2 sin 6 dQ = \a 2 . 

J e=o 

(3). Otherwise, immediately from the geometry of the figure, 



as in (i). 



/ 

x=o t/ Z=o 



dz dx 



yax — x 2 



*££*- 



V* 



3. Find the surface of the sphere intercepted by the cylinder in Ex. 2. 
From the figure, 

a a 

(I). sec y = - 



\S 



y/d> - x 2 - y 2 

P \x=a f'y- yax-X 1 dy dx 

J x =o Jy=° \/a 2 -x*-y 2 ' 

Integrate directly, or put sin 2 = x/(a -f- x) and integrate 
Hence S = 2a\it — 2). 
(2). Again, 

\S = / pQ sec y dp. 

«'o 

p — a cos b = a sin y. .-. y = \ic — Q 
$S = — a 2 fo cos dQ = — a 2 [Q sin -f cos 6]*" = - a\\ir — I) 

Lengths of Curves in Space. 

287. As in plane curves, the length of a curve in space is defined 
to be the limit to which converges the sum of the lengths of the sides 
of a polygonal line inscribed in the curve. 



Art. 287. j APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 405 






Since \~£) A ** = Aa * + Ay2 + **» 



with similar values for the derivatives —, — , 



'./N'-*.®"*®'* 



with corresponding values for s when >■ or 2 is taken as the indepen- 
dent variable. 

If the coordinates of a point on the curve are given in terms of a 
variable /, then 



and 



(D:= p* |j+ (if- 



EXAMPLES. 

1. Find the length of the helix 



x = a cos - , y = a sin - , 


measured from z = o. 

Take z as the independent variable. Then 

dx a . 2 dy a z 

- = -7-sin-, — =—cos — ; 
dz b b 1 dz b b' 






2. Find the length, measured from the origin, of the curve 

2ay — x 2 , bd l z = x z . 

'-/(«-^i)^=/('+=h=- +^-*+* 

3. Show that the length, measured from the origin, of 

y = a sin #, 42 = a 2 (.*r + cos x sin x), 
is x -\- z. 

4. Find the length of 

_ 2 !x 3 

y — 2 |/a* — *, ^ = # — — a(~» 
measured from the origin. ^»^- J = •* -\-y — *• 



4©6 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVII. 

5. Find the length, measured from the horizontal plane, of the curve 

x 2 y 2 /- --' 

288. Observations on Multiple Integrals. — The problem of 
integration always reduces ultimately to the irreducible integral 

dE being the element of the subject to be integrated. Or this may 
be taken as the starting-point and considered as the simplest element- 
ary statement of the problem for solution. This, in simple cases, may 
be evaluated directly, otherwise it may be necessary to integrate par- 
tially two or more times with respect to the different variables which 
enter the problem. There may be several different ways in which the 
elements can be summed. A careful study of the problem in each 
particular case should be made in order to determine the best way of 
effecting the partial summations, with respect to the limits at eacli 
stage of the process. 

One is at perfect liberty to take the elements of integration in 
geometrical problems in any way and of any shape one chooses, as the 
limit of the sum is independent of the manner in which the subdivision 
is made (see Appendix). This should be verified by working the 
same problem in several different ways. 

The applications of multiple integration in mechanics are numerous 
and extensive. Further application beyond the elementary geometrical 
ones given here is outside the scope of the present work. 

EXERCISES. 

In these exercises the results should be obtained by double and triple integra- 
tion, and also by single integration whenever it is possible. 

1. Find the volume bounded by the surfaces 

x 2 -f- y 2 = a 2 , z = O, z = x tan a. 



pa p Ya* - x* px tan a 

Ans. 2 / / / dz dy ax = $a z tan a. 

Jo «/o Jo 

2. Find the volume bounded by the plane 2 — 0, the cylinder 

(x- a) 2 + (y -bf = A' 2 , 
and the hyperbolic paraboloid xy = cz. Ans. it — R 2 . 

3. Find the volume bounded by the sphere and cylinders 

x i + y i ± z i - a 2 , x 2 + y 2 = b\ p 2 = a 2 cos 2 + b 2 sin 2 0. 

Ans. $(16 - 37t)(a 2 - b 2 )l 



Art. 



2SS.J APPLICATION OF DOUBLE AND TRIPLE ENTEGRALS. 407 



4. A sphere is cut by a right cylinder whose surface passes through the center 
of the sphere ; the radius of the cylinder is one half that of the sphere rt. Find the 
volume common to both surfaces. Ans. \{7t — 4 )a 3 . 

5. Show that the volume included within the surface 

^(-v'.-) = °- 

\a b c J 
is abc times the volume of the surface 

F(x, y, z) = o. 

6. Show that the volume of the solid bounded by the surfaces 

z = o, x 2 -\-y 2 = ^az, ^-fjj/ 2 = 2nr, is \nc^/a. 

7. Find the entire volume bounded by the positive sides of the three coordinate 
planes and the surface 

8. Find the volume bounded by the surface 

**+}>*+** = a*. Ans 

9. Find the volume of the surface 



abc 
90' 



QMOMt)*- ** *-* 



10. Show that the volume included between the surface of the hyperboloid of 
one sheet, its asymptotic cone, and two planes parallel to that of the real axes is 
proportional to the distance between those planes. 

11. Find the whole volume of the solid 

x i/a* + y 2 /b 2 + z'/c* = 1. Ans. \itabc. 

12. Find the whole volume of the solid bounded by 

{x 2 _j_y _j_ ^2)3 _ 2ja s xyz. Ans. |<z 3 . 

13. Use § 285 to show that the volume of the torus 

( X 2 _f_ y* 4. 2 2 _|_ C 1 _ a 2^ _ 4 ^2^2 _|_ ^2) j s 2 7t 2 Ca 2 . 

14. Find the volume of the solid bounded by the planes x = o, y = o, the sur- 
face (x -\- yf = 4az, and the tangent plane to the surface at any point/, g, h. 

Ans. \ah 2 . 

15. Show that the surfaces y 2 -\- z 2 = q.ax, and x — z = a, include a volume 
STtaK 

16. Show that the volume included between the plane = 0, the cylinder 
c — x 2 , and a paraboloid ax 2 -\- by 2 = 2z is |7T^(5« _I -j- b~ l ). 

17. Show that the whole volume of the surface whose equation is 

(x 2 -\-y 2 + z<1 f = cx y z * s equal to fi/ifio. 

18. Show that the volume included between the planes/ = ± k and the surface 

a?x 2 + £ 2 2 = 2(ax -f- ^s)j' 2 is \itJ^/^ab. 

19. Find the form of the surface whose equation is 

( x 2 /a 2 _|_ /2 /£2 _|_ ^2)2 _ x %j a 7. _|_ ^2/^2 _ ^2/^ 

and show that the volume is it 2 abc/\ ^2. 



4oS INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVII. 

20. Find the entire surface of the groin, the solid common to two equal cylin- 
ders of revolution whose axes intersect at right angles. 

Ans. i6A" 2 . /t being the radius of the cylinders. 

21. Find the area of the surface 

z' 1 -f- (x cos a -\- y sin a) 2 = a- 
in the first octant. Ans. 2a 2 esc 2a. 

22. Find the volume of the solid in the first octant bounded by xy = az and 

x -\- y -f- z = a. Ans. (\^ — log 4)tf 3 . 

23. Find the surface of the sphere x 2 -\- y 2 -f- z 2 = a 2 in the first octant inter- 
cepted between the planes x = o, y = o, x ■= b, y — b. 

b b 2 \ 



Ans. 



I , . b . b 2 \ 

a I 2b sin-i — a sin-' — ). 

y ■ 4/a 2 - b 2 ^ - b 2 ) 



24. A curve is traced on a sphere so that its tangent makes always a constant 
angle with a fixed plane. Find its length from cusp to cusp. 






CHAPTER XXXVIII. 
INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS. 

289. Classification. — A differential equation is an equation which 
involves derivatives or differentials. 

An ordinary differential equation is one in which the derivatives 
are taken with respect to one independent variable. These are the 
only kind that we shall consider. 

Differential equations are classified according to the order and 
degree of the equation. The order of a differential equation is the 
order of the highest derivative contained in the equation. The degree 
of the equation is the highest power of the highest derivative involved. 

290. We shall consider in this text only examples of ordinary dif- 
ferential equations of the first and second degree in the first order, 
and a few particular cases of the first degree in the second order. 

291. Examples of Equations of the First Order and First 
Degree. — The derivative equations of the first order and first degree 

dy dy 2 „ dy dy 

— = cos x, 2X ^— = 7y — xy, ax z yr - - = 2x y, 

dx ' dx ^ ■ J dx dx "" 

when multiplied by dx, are equivalent to the differential equations of 

the first order and first degree 

dy = cos x dx, 2X dy = (3 V — xy)dx, ax 2 j^dy — 2X dy —y dx. 

dy 
In general, any linear function of—, 

in which and ip are constants, or functions of x ory, or of x and y, 
is a derivative equation of the first degree and order. When multi- 
plied by dx it becomes the general differential equation of the first 
degree and order 

dy -\- ip dx = o. 

292. Examples of Equations of the First Order and Second 
Degree. — The equations 

f dy\ 2 (dy\ 2 (dy 



&; - *[&) ~ 2y \T x ) + ax = °- 

are of the second degree and first order. Written differentially, 
dy 2 = ax 3 dx 2 , x dy 2 — 2y dy dx -\- ax dx 2 = o. 

409 



4io INTEGRATION FOR MORE THAN ONE VARIABLE. [Cm. XXX VIII. 



In general, the type of an equation of the first order and second 
degree is 



*(S) + <£)+ 



where 0, ?/•, ^ are functions of x f y or _r andj^, or constants. 

293. Equations of the Second Order and First Degree. — Such 
equations as 

d 2 y _ d 2 y dy 

y m ~ (£) =^ l0 s-"> *> £ + {* £ -')■ = °» 

are of the second order and first degree. 

294. Solution of a Differential Equation. — To solve a given 
differential equation 

*\x,y>y') = °> 

dy 
wherejy = — , is to find the values x and^ which satisfy the equa- 
tion. Thus, if the values of x and y which satisfy the equation 

0O~, y) = © 

satisfy a differential equation F = 0, then = o is a solution of 
^= o. 

The solution of a given differential equation may be a particular 
solution or it may be the general solution. The general solution in- 
cludes all the particular solutions. Or the solution may be a singular 
solution, which is not included in the general solution. The complete 
solution of a differential equation includes the general solution and the 
singular solution. The meaning of these solutions will be developed 
in what follows. 

The solution of a differential equation is considered as having 
been effected when it has been reduced to an equation in integrals, 
whether the actual integrations can be effected in finite terms or not. 

Equations of the First Degree and First Order. 

295. The simplest type of an ordinary differential equation of the 
first order and degree is 

dy=/(x)dx. (1) 

Integrating, we obtain the solution 

y = F( x ) + c, (2) 

where F(x) is a primitive of/(x) and c is an arbitrary constant. For 
a particular assigned value of c, (2) is a particular solution of (1), 



Art. 295.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 411 

and is the equation of a particular curve in a definite position. At 
each point of the curve (2), 

is the slope, or direction of the curve (2). For different values of 
ewe have different curves. The ordinates of any two such curves 
differ by a constant. Equation (2) is then the equation of a family 
of curves having the arbitrary parameter c. This singly infinite sys- 
tem of curves, or family of curves with a single parameter, is the 
general solution of the differential equation (1). 

296. Every equation of the first order and first degree can be 
written 

Mdx -f- N dy = o, (1) 

where, as has been said before, MandJVare either constants, functions 
of a: or r, or functions of x andj'. 

297. Solution by Separation of the Variables.— This solution 
consists in arranging the equation 

Mdx + Ndy = o, (1) 

so that it takes the form 

(p(x)dx + ip(y)dy = 0. (2) 

The process by which this is effected is called separation of the 
variables. When the variables have been thus separated the solution 
is obtained by direct integration. Thus, integrating (2), 

fcP(x) dx + fip(y) dy = c, 

where c is an arbitrary constant, and is the parameter of the family 
of curves representing the solution. 

I. Variables Separated by Inspection. — A considerable number of 
simple equations can be solved directly by an obvious separation of 
the variables. The process is best illustrated by examples which 
follow. 

EXAMPLES. 

1. Find the curve whose slope to the jr-axis is — x/y, and which passes through 
the point 2, 3. 

The geometrical conditions give rise to the differential equation 

& x 1 1 j 

- — = — — , or y ay -f- x dx = o. 

ax y 

The solution of which, obtained by integration, is the family of circles 

x 2 + y = c 2 . 

The particular curve of the family through 2, 3 is 
x 2 + y 2 = 13. 



412 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII. 

2. Find the line whose slope is constant. 

- = m gives the family of parallel straight lines y = mx -f- c. 

3. Find the curves whose differential equation is 

x dy -f- y dx = o. 
The variables when separated give 

dx . dy 

\. 4~ = o. 

x y 

.'. log* -f- log y = c, or xy = k. 

Otherwise we may write the solution xy — e c . This is a family of hyperbolae 
having for asymptotes the coordinate axes. 

If we observe that x dy -\- y dx is nothing more than d(xy), the solution xy = c 
is obvious. 

4. Find the curve whose slope at any point is equal to the ordinate at the point. 

tt d y d ? j 

Here -y- = y. .-. -- = dx. 

dx y 

Hence \ogy = x -{- c , or y = e x + c = e c e x = ae x , 

which is the exponential family of curves. 

5. Find the curve whose slope is proportional to the abscissa. 

Ans. The family of parabolae jk = ax 2 -f- c, in which c. the constant of integra- 
tion, is the parameter. 

6. Find the curve whose slope at x, y is equal to xy. Ans. y = c e ^ x • 

7. Find the curve whose subtangent is proportional to the abscissa of the point 
of contact. 

dx dx dv 

Here y — - = ax. . • . — = .a — gives 

' dy x y & 

log x = a log y -(- c, or y a == kx. 

8. Find the curve whose subnormal is constant. 

y - — a gives y- = 2.ax -\- c, the parabola. 

X 

9. Find the curve whose subtangent is constant. Ans. y — ce a . 

10. Find the curve whose subnormal is proportional to the «th power of the 
ordinate. What is the curve when n is 2 ? 

11. Find the curve whose normal. length is constant. 

Here the geometrical conditions give the differential equation 



■J- ♦■*)■- 



y-x lJ r ;;- --*■ ••• <** = 



\/a--y* 

Integrating, x — c = — (tf 2 — y 2 )^, or the family of circles 

{x-c) 2 +y 2 = a\ 

with radius a. having their centers on the .r-axis. 

12. Find the curve in which the perpendicular on the tangent drawn from the 
foot of the ordinate of the point of contact is constant and equal to a. 






Art. 297.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 413 

The differential equation of condition is 



++®' 



°t= = dx. 



v/> 



The solution is therefore the family of curves 

c + x - a log (y + \f y 2 - a 2 ). 
When c = o this is the catenary with Oy as axis. 

13. Find the curve in which the subtangent is proportional to the subnormal. 

14. Determine the curve in which the length of the arc measured from a fixed 
point to any point Pis proportional to (1) the abscissa, (2) the square of the 
abscissa, (3) the square root of the abscissa of the point P. 

(i). A straight line. 



(2). The condition is 



ds 2 = dx 2 + dy 2 = ^ dx* 



a dy = |/r 2 — a 2 dx. 
The solution of this is 

c -f- ay = \x \/x 2 - a 2 - \a 2 log \x + y ' x 2 - a 2 ] , 
(3). The geometrical condition can be written s = 2 \/ax. 



ds — ^ — dx. dx 2 -f dy 2 = ds 2 = —dx gives 

Ax x 



dy 



•j ! 






Put x = z 1 and integrate. The result is the cycloid 



c -j- y = ^x{a — x) + a sin- 1 



\a 






Ex. 14, really leads to a differential equation of the first order and second 
degree, which furnishes two solutions which are the same. 

15. Find the curve in which the polar subnormal is proportional to (1) the radius 
vector, (2) to the sine of the vectorial angle, (i). p — ce a $. (2). ft = c — a cos 0. 

16. Find the curve in which the polar subtangent is proportional to the length 
of the radius vector, and also that curve in which the polar subtangent and polar 
sub-normal are in constant ratio. Ans. ft = cent. 

17. Determine the curve in which the angle between the radius vector and the 
tangent is one half the vectorial angle. Ans. p = c(i — cos 0). 

18. Determine the curve such that the area bounded by the axes, the curve, and 
any ordinate is proportional to that ordinate. 

X 

If fl is the area, £1 - ay. . •. d£l — y dx — a dy. . : y = «"! 

19. Determine the curve such that the area bounded by the x axis, the curve, 

and two ordinates is proportional to the arc between two ordinates. 

dy 
£1 = as. .•. y dx = a ds, dx = a , . 



414 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII. 

This gives, on integration, the catenary 

c + x = a log (y + \/) ri ~ <)• 

20. Find the curve in which the square of the slope of the tangent is equal to 
the slope of the radius vector to the point of contact. 

The parabola x* -\- y^ — £*, or (x — yf — 2c{x -f- y) -f c 2 = O. 

21. Solve M dx -f- N dy, when Mx ± Ny = o. 
(i). 7l/r -f- A> = o gives M/N = — y/x. 

Substituting in the equation, — - = — . . •. x = cy. 

(2). Mx — Ny — o gives M/N — y/x. 

dx dy 
Substituting in the equation, (- — = o. . •. xy = c. 

II. Solution when the Equation is homogeneous in x and y. — When 
the equation 

Mix -f N dy = o 

is such that M = (p(x, y), N = ip(x, y) are homogeneous functions 
of x and^ and of the same degree, the solution can be obtained by 
the substitution^ = zx. 
We have 

N - >Hx,y) ~ w> 

Divide the numerator and denominator by x n , n being the degree 
of <p or ij\ 

d y i dz tpt \ 
■■■ di= z + x Tx=- F ^- 

Hence 

dx dz 



x ' z + F{z) 

and the variables are separated. The integration of this gives an 
equation in x and z. On substituting y/x for z the solution of the 
original equation is obtained. 

EXAMPLES. 

1. Solve the equation (2x 2 — y 2 )dy — 2xy dx = O. 
dz 2s 



Put y = zx. .'. z 4- x — = 

\L — Z- 

9 _ *2 

dz. 



dx 2 - 2 2 ' 

ate _ 2 - z 2 
Integrating, 



log x — c — log z. 

Replacing z by y/x, we have 

x* =y\c - logy). 



Art. 298.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 415 

2. Determine the curve in which the perpendicular from the origin on the 
tangent is equal to the abscissa of the point of contact. 

Arts. Thr circles .v- -\- j ■'-' = lex. 

3. Find the curve in which the intercept of the normal on the x-axis is propor- 
tional to the ordinate of the point of contact. 

x -\-y c - ~ — my. . •. (x — my)dx -f y dy = 0, etc. 

4. Find the curve in which the subnormal is equal to the sum of the abscissa 
and radius vector. 

5. Find the curve whose slope at any point is equal to the ratio of the arith- 
metic to the geometric mean of the coordinates of the point. 

6. Solve y 2 dx -f- (xy + x 2 ,dy = o. Ans. xy 2 — c\x -f 2y). 

y x* 

7. Solve x 2 y dx = (x* -\-y z )dy. Ans. log- = . 

298. Solution when M and N are of the First Degree. — The 

equation 

(a x x + b x y + c x )dx = (a 2 x + b 2 y + c 2 )dy (1) 

can always be solved as follows : 

Put x — x' -\- h, y = y' -J- k, where h and k are arbitrary 
constants. Then (1) becomes 

dy' a x x' -\- b x y' -f- a x h -f- b x k -j- c x 



dx' a 2 x' -f b 2 y' -\- a 2 h -j- b 2 k -f- c 2 ' 
I, If a x b 2 =£ aj> x , assign to h, k the values which satisfy 



to 



a x h + bjz + c, = o, [ , ^ 



a 2 A + ^ + c 2 = o. j" 
Then (2) becomes 

dy' _ ^7 r f / -f- ^y 
dx' a 2 x' -f 3^' ' 

This is homogeneous and can be solved by § 297. 
\{f{x',y') = o is the solution of (4), then/(^ — h,y — k) 
is the solution of (1). 

II. Uab = ah, let - 2 = ^ = fl*. 



(4) 



Then (1) becomes 



a x b x 



dy _ a x x + b x y + e x ^ 



dx m(a x x -\- b x y) -f- £ 2 

Put = 0^ -\- b x y. Then. (5) becomes 

dz , . z -\- c. 

0, 



dx 1 x ??iz -\- c 2 

in which the variables can be readily separated. 



41 6 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cn. XXXVIII. 



EXAMPLES. 

1. Solve (37 - 7x +7)dx + {jy - 3* + 3)dy = o. 

Ans. (y -x+ i)\y + x - if = c. 

2. Solve (2x -\-y + i)dx -f- (4* + *y — t)dy — o. 

Ans. x -\- 2y -\- log(2x -\-.y — i) = c. 

3. Solve. ( 7 j + x + 2)a& - (3* + 5/ + 6)4> = o. 

Ans. x-)-5;-f 2=4x-;| 2) 4 . 

299. The Exact Differential Equation. — The differential equation 

Mdx-\-Ndy = o 

is said to be an exact differential equation when it is the immediate result 
of differentiating an implicit function /"(or, _>') = o. 
In fact, if 

u =f(x,y) = o, 

then </w = -^-</.r 4- ^-^ = o 

ox by 

gives an exact differential equation. 

300. Condition that M dx -\- Ndy = o be Exact. — Since M must 
be the first partial derivative with respect to x, and .A 7 the first partial 
derivative with respect to y of some function/"^, y), then 

dx dy 

But since 

dy ey 

dy dx dx dy* 

we must have the relation 

dM _dN 

~dy~~~dx' '*' 

existing between M and N in order that Mdx -f- Ndy = o shall be 
exact. This condition is also sufficient, and when (1) is satisfied 
Mdx -f- Ndy is an exact differential. 

For,* let V= j Mdx. 

9T_ d 2 V _ dM _ dN 

•"" ay - ' a7a7 ~ 6v~ ~ dx ' 
dx d*v 



dx dx dy 



— ^/^(^ = F + ^ 



This is due to Professor James McMahon. 



Art. 301.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 417 

where the constant of integration (f)'(y) is some function of;' or a 
constant independent of x. Therefore 

Mdx + Ndy = Jjp.v + ^-Ldy + <fi'(y)dy, 

= d[V +</.(.))], 
an exact differential. 

301. Solution of the Exact Equation. 

rr BM dN , 

11 —z — = 5— , there exists a function «, of a- and_>', such that 

du — M dx -\- N dy. (1) 

Since M=. — , il/ contains the derivatives of only those terms in 

u which contain a. Integrating (1) with respect to x (y being con- 
stant), we have 

u=fMdx+<p(y), ( 2 ) 

where <p(y) represents the terms in u which do not contain x. 
To find (p(y), differentiate (2) with respect toy. 

. •. a = N = ^ / Mdx + -^-. 
Hence 



9>> 9y 



yw (3) 



As was said, is independent of x and so also is ^— , as is verified 

ay 

by differentiating (3) with respect to x; 

JL I # - — /V</* 1 - — - — - 

9a ( dy J ^ ) dx d y 

Integrate (3) with respect toy. 

■ •• 000 = f\ N -%y-f Mdx } & + c - 

Therefore the solution of (1) is 

u = C Mdx + / \n-§~ f Mdx I dy + c = o. (4) 
In like manner, working first with N instead of M, 

J Ndy +f j M- ^f^dy J dx + c = o (5) 

is also a solution of (i). 



4i3 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cn. XXXVIII. 

302. Rule for Solving the Exact Equation. 

/ Mdx contains all the terms of the primitive containing^". Also, 

since N — - — / M dx is independent of x, - — / Mdx must contain 
dx J dxj 

those terms in JV containing x. Therefore to obtain 

integrate only those terms in N which do not contain x. Hence the 
rule. Integrate M dx as My were constant; integrate those terms in 
N dy which do not contain x; equate the sum of these integrals to a 
constant. 

A like rule follows for effecting (5), § 301. 

EXAMPLES. 

1. Solve (3^ 2 — \xy — zy*)dx -f- (3_y 2 — \xy — 2x 2 )dy = o. 
Here _ = _ 4 *_ 4 , = __. 



(Mdx = *? - 2x 2 y - 2xy 2 ; J 2>y 2 dy = 



r 



Therefore the solution is 

x 3 — 2x 2 y — 2xy 2 -f- y 3 =. c. 

2. Solve (x 2 + y 2 )(x dx + y dy) -f x dy - y dx = o. 

x i _L y2 y 

Ans. L ^- + tan-i — — c. 

2 ' x 

3. Solve (a 2 + $xy - 2y 2 )dx 4. {ix—yfdy - o. 

Ans. a' 2 x -)- $y A — 2xy 2 -\- ^x 2 y = c. 

4. Solve (2ax -f- by 4- g)dx 4- (2cy -|- bx + e)dy — o. 

Ans. ax 2 -\- bxy -f- 9' 2 4" <£"■* + 9' — &' 

5. Solve (w dx 4- «<^) sin (oti 4~ n )') — ( n l ^ x "f m d)') cos («x 4- my). 

Ans. cos (w,r 4- «»') 4" s i n ( w -*" + m )') = f - 

6. Solve 2x(jc + 2j)^x -j- (2x 2 — y 2 )dy = o. 

Ans. x' A 4" 3 a " 2 J' — V s = c - 

303. Non-Exact Equations of the First Order and Degree. — We 
have seen that when a primitive equation f[x,y) = o is differentiated 
there results the exact differential equation <p[x,y,y') = o, writing 
y' for the derivative of r with respect to x. 

If now between f = o and (/> = o we eliminate any constant 
occurring in /and 0, we get another equation, */'(.v, r, r') = o, which 
is a differential equation satisfied at every point on /•=. o. Therefore 
/"= o is a primitive of ?/' — o. But //' = o will not be an exact 



Art. 303.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 419 

differential of the primitive/" = o, although/" = o is a solution of the 
differential equation tp = o. 

To fix the ideas, consider the equation 

ax -f- by -\- cxy -j- k = o. (1) 

The exact differential equation of (1) is 

(a -j- cy)dx -f- (b -J- cx)dy = o. (2) 

When (2) is integrated the constant of integration restores the 
parameter k of the family (1) and (1) is the solution of (2). That is 
to say, the family of curves (1) obtained by varying the parameter /£ 
gives the solution of the exact differential equation (2). 

The constant k was eliminated from (2) by the operation of differ- 
entiation and restored by the process of integration. 

Eliminate a between (1) and (2) by substituting 

a -J- cy — — = — 

X 

from (1) in (2). There results the differential equation ■ 
dy dx 



by + k x{b -U ex) ' 

1 dx c dx 



(3) 



or 



b x bb -\- ex 

bdy dx c dx 



by -|- k x ex + b' 
Integrating and adding the arbitrary constant — log c', 
log (by -|- k) -\- \og(cx -f b) — log x — log c' — o. 

{by -f- ^)(<^^ + ^) — c/jc > 
or (^c — c')jc -\- b 2 y -{- be xy -\- kb — o. 

Putting the arbitrary parameter in the form kc — c' = ab, this 
equation becomes the original primitive 

ax -j- by -\- cxy -\- k = o. 

This equation with the variable parameter a is the solution ot the 
differential equation (3). 

The differential equation (3), or 

(by -f- ^)^jc — x(b -\~ cx)dy — o (4) 

is not an exact equation, for 

A (by + k) = 5, A ( - bx -ex") = - b - 2C*. 

But (1) is the primitive of (3) as well as of (2). 

Again, if we eliminate first b and then c between (1) and (2), we 



420 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII. 

shall get two other differential equations, neither of which is exact, 
but each of which has (i) for solution with variable parameters b and c 
respectively. 

Observe particularly that if (4) be multiplied by i/x 2 , it becomes 
an exact differential, 

=^r- <* — & =°> (5) 

since 

1 l b y + k \ - d I b + cx \_b 
dy \ x 2 j ~~ dx \ x J x r 

Integrating this exact equation (5) under the rule §302, the 
solution is 

qx -J- by -f- cxy + k = o, 

the same equation as (1) with q for parameter. 

304. Integrating Factors. — In the preceding article we have seen 
that the same group of primitives can have a. number of different 
differential equations of the first order and degree. The form of any 
particular differential equation depending on the manner in which an 
arbitrary constant has been eliminated between the primitive and its 
exact differential equation. In the example above, when the differential 
equation was not exact, it was made exact by multiplying by 1/x 2 . 
Such a factor is called an integrating factor of the differential equation 
which it renders exact. 

The number of integrating factors for any equation 

Mdx + Ndy=o (1) 

is infinite. For, let /u be an integrating factor of (1). Then 
ju(Mdx -f- N dy) is an exact differential, say dn, and 

fi(M dx + N dy) — du. 

Multiply both sides of this equation by any integrable function of 
u, say /(a), 

fi/{u){Mdx -f- Ndy) =/(u)du. (2) 

The second member of (2) is an exact differential, and therefore 
also is the first. Hence, when /u is an integrating factor of (1), so 
also is /Jf(u), where f{u) is any arbitrary integrable function of u. 

In illustration consider the equation 

y dx — x dy = o. 

This is not exact, but when multiplied by either—,, — , or -- 

1 y 2 xv x~ 

it becomes exact and has for solution 

x 

— = constant. 

y 



Art. 305. J INTEGRATION OF DIFFERENTIAL EQUATIONS. 421 

The general solution of the differential equation 

M dx -\- N dy — o 

consists in finding an integrating factor jj. such that 

}A(Mdx + Ndy) = o 

is an exact differential, then integrating by the method given as the 
solution of the exact equation. 

The integrating factor always exists, but there is no known method 
by which it can be determined generally. The rules for determining 
an integrating factor for a few important equations will now be given. 

305. Rules for Integrating Factors. 

I. By Inspection. — While the process of finding an integrating 
factor by inspection does not, strictly speaking, constitute a rule, in the 
absence of a general law for finding the integrating factor it is an 
important method of procedure. An equation should always be ex- 
amined first with the view of being able to recognize a factor of inte- 
gration. The process is best illustrated by examples. 

EXAMPLES. 

1. Solve y dx — x dy + f(x)dx = o. 

The last term is exact; its product by any function of x is exact. Therefore 
any function of x that will make y dx — x dy exact is an integrating factor. Such 
a factor is obviously 1/x 2 . 

- ydx-xdy , Ax) dxt=2 



or 






'( 


x 2 ' ~X~ T 

3 +4?*-* 


gives 


the solution 


y_ 

X 


+f>=<- 


2. 


Solve 


y dx -\- log x 


dx = 


: x dy. 



Ans. ex -f y -f- log x -f I = O, 

3. Solve (1 + xy)y dx + (1 — xy)x dy — o. (Factor i/x 2 y 2 ). 

Ans. ex = ye x y. 

4. Integrate x a y&(ay dx -f bx dy)= o. 

Obviously x ka-i-ykb-i j s an integrating factor, where k is any number. 
On multiplying by the factor we get 

axka-\ykb dx -\- bxkaykb-\ dy = jd{xka-ykb s ) = o, 

the solution of which is evident. 

5. Integrate 

xay£{ay dx + bx dy) -f- x*\yt\(a x y dx -f b x x dy) = O. 



422 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII. 

The factors x ka — »— ^ykb— 1-£, x i l a l -i-a 1 ^k l i> 1 -i-^ i 
make the expressions 

x*yP(ay dx -f bx dy) and x a tyPi(a } y dx + b x x dy) 
exact differentials respectively, whatever be the values of the arbitrary numbers 
k and k v 

Therefore, if k and k x be determined so as to satisfy 
ka — i — a = k 1 a 1 — i — a v 
kb - i - p = k x b x - i - /5 V 

the factors are identical and these values of k and k x furnish the integrating factor 
of the equation proposed. 

6. Solve (y z — 2yx 2 )dx -f (2xy 2 — x' s )dy — o. 

Ans. x 2 y 2 (y 2 — x' 2 ) = c. 

7. Solve the equation 

[y + xf(x* + y 2 )]dx = [x - y/(x 2 + y 2 )]dy. (l) 

This is the differential equation of the group or family of rotations. Put 

x 1 _|_ y% _ ,,2. 

Rearranging (i), 

ydx-x dy -f /(r2) .{xdx + y dy) = o, 
2(7 djc — Jf </y) +/(^ 2 )^ 2 = o. 
This can be written 

(> dx - x dy) - (x dy - y dx) + f(r 2 )dr 2 = o, 

or y*d(j) -*d U) + f K r 2 )dr 2 = o. 

An integrating factor is obviously ^ 2 . Whence 

x- y 1 r 2 

Integrating, 

tan-.*-ta„-i£ + /*-*S** = ,. 

7 x J r l 

II. Whenever an integrating factor exists which is a function of x 
only or ofy only, it can be found. 

Making use of the fact that e z is always a factor of its derivative: 

(a). Let 2 be a function of x. 

In e 2 (Mdx -f iV^) = o, 

put -flf ' = *W, -AT' = *W. 

Then __-^_ -j =**A +^ 

dy 9>/ ox ax dx 

The condition that <f shall be an integrating factor is 

, dM xAT dz , 3JV 

e '-8y-= eZN dx- + eZ 6x-> 

dM _dN 

dy dx 
or dz = — dx. 

N 



d y - 



Art. 305.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 423 

If, therefore, 



dM _ dN 



cP{x) 






N 
is a function of x only, then 

z = / <p(x)dx. 

Hence e J ' is an integrating factor of M dx -f- N dy = o 
whenever 

dy dx ( 1 ) 

is a function, (p(x), of .r only. 

(3). In like manner, letting e z be a function of v only, we find 

fo(y)dy , 

that e J is an integrating factor of 

M dx -f N dy — o 
when 

cW aJSf 

^~~^~ ( 2 ) 

is a function, ip(y), ofy only. 

(c). Whenever the expression (1), (2), or (p(x), ip(y) is constant, 
then e* or e y , respectively, is the integrating factor. 

EXAMPLES. 

1. One of the most important equations under this head is Leibnitz's linear 
equation, 

% + * = « 

where /'and Q are functions of x or are constants. 
This equation, {Py — Q)dx -f dy — o, is such that 

dM dN 

Jy ?f__ d 

N ~ ' 

fPdx 

Therefore it has the integrating factor e J 

ef PdX {dy + Py dx) = e' Pd * Q dx. (2) 

f pdx f pdx ^ , ,( f Pdx \ 

Since e J dy -f e J Py dx - d\ye J ), 

on integrating (2), 

, = rJ > »\fJr"Q* + *\. (3) 

This is the solution of the linear equation (1). 



424 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch, XXXVIII. 

2. Bernoulli's Equation. — The equation known as Bernoulli's 

I + * = <*"• 

in which P, Q are functions of x or are constants, reduces to Leibnitz's linear equa- 
tion. For, multiply by ( — n -f- i)/y n , and put v = y~ n + l . Equation (i) becomes 

*L + (I _ n)Pv = (I - n)Q, 
which is linear in v. 

3. Solve f\y) -^ +Pf(}') = Q, where P, Q are functions of x. 
Put v = f(y)- The equation becomes 



*£ + Pv = Q > 



which is linear in v. 



III. When Mx ± Ny ^ o, //^re #r£ two cases in which the inte- 
grating factor of M dx -\- N dy = o c#« <$<? assigned. 

(i). When vT/, TV are homogeneous and of the same degree, then 

— =- is an integrating factor. 

Mx + Ny 6 & 

(2). When .#/", N are such functions as 

M = ycp(x xy), N = xip(x x y), 

then -=-= ^v- is an integrating factor. 

Mx — Ay & & 

Proof: We have the identity 

Mdx + Ndy 

= i j ( JK* + 7Vy)</ log (*y) + (Mx - Ny)d log (*/.y) } . 
(1). Divide by Mx + Ny. 
Mdx + Ndy ... . . . .J/* - i\Y ,. /*\ 

Mx+Ny = * rf l0g ^ + - M7TW <n ° g b)' 

= . Id \og(xy) + l/(^y log (j), 
if M, N are homogeneous functions in x, y and of the same degree. 

X 

log — 

Since x/y — e y , this can be written 

»g±** = Vlog (*> + *■ (log ^.OgQ, 

where tt = log (xy), v — log (x/y). 



Art. 305. J INTEGRATION OF DIFFERENTIAL EQUATIONS. 425 

This case is otherwise solved by the substitution y = zx. see 
§297,11. 

(2). Divide by Mx - Ny 

M dx 4- N dy 1 Mx -f Ny 

Mx-Nv' = ^ Mx^fy °* ^ + ^ l0 ^ (*/»■ 

If M=y<p(xy), N = xt/>(xy), then 

7J/ air 4- A T </y 

. •. j^I^y - = *A*y)<? log (*y) + u i og (*/,), 

= i^(log xy)d log xy + Ji/ log (x/y), 
= \F{u)du + ^jfc. 

Writing as before, xy = £ log * y , « = log xy, v = log x/y. 

(3). The cases in which Mx ± Ny = o were solved in § 297, I, 
Ex. 21. 

EXAMPLES. 

Solve by integrating factors the following equations: 

1. y dx — x dy -(- log .r </.r — o. (I, Ex. I.) 

^tw. ex -\- y -\- log jr -(- 1 = o. 

2. #(•* ^ -(- 2y dx) = xy dy. Ans. a log x' l y = y -\- e. 

3. (x 2 + 2xy - y 2 )dx = (x 2 - 2xy - y 2 )dy. Ans. x 2 -f y 2 = e(x + y). 

l m ^ + ± = J d l_ d A. Ans.x 2 -y 2 + xy=e. 

x y \x y J 

5. (x 2 y 2 -f- xy)y dx + {x 2 y 2 — 1) x dy — o. Ans. y — ee*y. 

6. (x 3 y* -f- i)(* ^ + ^ <*r) + (xy + xy)(^ rf* - x dy) = o. 

^«j-. xy ■ = log ey 2 . 

7. x 3 dx -f(3.r 2 / -f- 2y*)dy = o. ^w. * 2 -\- 2y 2 = c \/x 2 4- /'. 
8- (>> + J V'j^)^- — (.r -f .r i / ^> / )'^ / == o. Ans. y = ex. 
9. (x 2 + j/ 2 -f- 2*)<&r 4- 2y ^' = O. ^«J. .r 2 4- y 2 = ce- x . 

10. (3-r 2 - y 2 )dy = 2xy dx. Ans. x 2 - y 2 - cy\ 

11. 2xy dy = (x 2 4- y 2 )dx. Ans. x 2 _ j 2 - ^ 

x y' s 

12. (x 2 / — 2xy 2 )dx — (x 3 — 3x 2 /)^v. ^«i. h log^ = c. 

y x 

13. (3^ 2 J 4 4- 2xy)dx = (.r 2 — 2x 3 y 3 )dy. Ans. x*y* -\- x 2 — <y. 

14. (jv* 4- 2/)^c 4- (.ry 3 4- 2/ 4 — 4x)dy = o. Ans. xy -\- y 2 -\- 2x/y 2 = e. 

15. (2x 2 y — zy*)dx 4- (3^ 4- 2xy 3 )dy — o. 

36 7* 10 IS 

Ans. $x~™y~™ — I2x~"y~™ = e. 

16. {y 2 4- 2^- 2 /)^ + ( 2 * 3 - ; 9')^ / = °- Ans - 6 ^ = ^" 3 J^ ? 4- ft 



426 INTEGRATION FUR MORE THAN ONE VARIABLE. [Ch. XXXVIII. 

17. x 4- — ay = x 4- I. y2«.r. v = V- cx a . 

dx I — a a 



18. (I + x 2 )</>' = {in -f- ;ri')^r. ^//.r. y — mx -|- <; |/i 4- x 2 . 

^ </>' tan v , . _, .x 3 — -zx -\- c 

19. 4-H — = (•* — J ) sec y> Ans. sin i' = ^ ! . 

dx ^ x 4- I V 3(* + I) 

20. - ; ■ — x — y. Ans. y = x — i 4- a?-*. 

21. '-'- + J = ^J' 3 - -<4«J. — = x 4- £ + c* 2 *. 

/v 1' 

22. -— = //— 4- e*x*, Ans. y = x n (e x 4- c). 
dx x 

23. *+l=i£, = I . A,.4 = i + «=-. 

306. Solution by Differentiation. — A number of equations can 
be solved, by means of differentiation as equations of the first order 
and degree. 

EXAMPLES. 

1. Let i> = -f- . Let the differential equation be 

dx 

x=f(p). (1) 

Differentiating with respect to/, 

dx=f\p)dp. 
Since dy = pdx, this gives the equation 

dy=f\P)pdp. 

• •• y =jf\P)pdpj r c. (2) 

The elimination of/ between (1) and (2) gives the solution. 

2. In like manner, if the differential equation is 

y=AP)* (*) 

on differentiation we have 

dy =/'(/>) dp. 

... pdx=f\p)dp, 

d*=nfdp. 
... x= f£fdp+c. w 

The elimination of/ between (1) and (2) is the general solution of (1). 

3. x = jp + log /. 

^;w. x + I = ± |/2j 4- r 4- log ( - 1 ± \/2y 4- c). 

4. x 2 / 2 = 1 4- p\ 

Ans. e 2 y 4- 2cxey -\- c 2 = o. 

5. ;/ = «/ + ty 2 . 

/*»j. x± \fa 2 4- 4<Jy = a log (a ± |/« 2 4- ^by) 4- r. 



Art. 306.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 427 

EXERCISES. 

1. $e* tan,r dx -f (1 - e*) sec 2 ^ dy = o. Ans. tan;' = <i - e*f. 

2. (4J* + 3*)^ + <y - **)dx = o. 

2V - (i - Vi )* 

3. (2* - y + i)dx + {x+y - 2)dy = o. 

.4/2 (3^ — 1) 
,4/m. log ;2( 3 x - i) 2 + (3y - 5) 2 { = V2 tan-i - 3j/ _ 5 - + * 

4. ( A -3^ _ 2 mxy 2 )dx + 2w.x 2 >' </y = o. Ans. x 2 e x + wy 2 = or 2 

5. j'(2x/ + e*)dx - t 'M' = o. Ans. x 2 y + ** = cy 

6. ^/ 4- (> - e-*)dx = o. ^'"- ye* = x 4- c 

7. cos 2 x </y + (y - tan *)</* = o. . ^«J. v - ce-*»** = tan * - 1 
S. ( x 4- i)dy = «y </jc + '*(■* + W+I ^- ^" J - > = (*" + «■)(•*■ + 0" 

b sin x 4" cos •* 
9. </j' = (by 4- a sin *)<**. ^«J. y = ce** — a - - — ^ 2 - 

10. * - -2_ = (, 4- I)'. ^«J. 27 = (x 4- 1)* + c{x 4- i) 2 
d£c ;c 4~ I 

11. xdy = ny dx 4- e*x*+*dx. Ans. y = x«{e* + c) 

12. dy = (^ 4" 1)* <**• ^ J = "^ " ' 

13. cos ^j+j sin * 4r = dx. Ans. y = sin x 4- r cos ^ 

14. 41 - * 2 ) — 4- (2* 2 - 1)7 = **?• Ans - y = ax + ** ^ ~ x ' 2 

■* + / , ^ + c 

15. (* + ;0 2 4' = « 2 <**•" ^ —T~ = tan "V 

jr — ^ 4- a * 



16. (x -yfdy = a? dx. Ans. log x _ y _ a = 2 

17 . x 2 dy + (/ _ 2^7 - * 2 K-r = o. ^«J- / = x 2 \i 4- 'W 

, 8 . ,*+£=,. ^«.« = ^ 

1Q ( X 2 I «2 _ fl 2W dx 4_ (^ _ y _ ^2) j, ^ = O. 

1 ^ ^. x* - y* 4- 2^!>' 2 - 2a 2 * 2 - 2/, 2 ;- 2 = c. 

20. dy = (xY - l)*y dx. Ans. y*{x> + 1 4" «**) = "• 

21. 2x7 ^ 4- (j 2 - * 2 ) d y = °- j^_ s - f + x2 = cy ' 

11. (x 4- y)dy +{*- y¥* = o- ^ w - lo s ^ 2 + y ' + tan_I ^" = £ ' 

23. (xy 4- x 2 / 2 4- *y + J)/ ^ + (^ 3 " • r2 >' 2 ~ ^ + I)x ^ J ' = °* 

Ans. x 2 ;' 2 — 2jtj/ log cy — I. 

OA ^. -L ; i- v - - ^ ;/J - xn y - axJ r c - 

'*' dx ^ x y x n ' 

25 ' ^ - * 'dx' \ I+ dx) K l + ** 



CHAPTER XXXIX. 

EXAMPLES OF EQUATIONS OF THE FIRST ORDER AND 
SECOND DEGREE. 

307. The equation of the First order and Second degree is a 

dy 

quadratic equation in — of the form 

where A, Bare, in general, functions of xa.ndy. 

dy 

We shall represent-^- by p. Equation (1) can be written symboli- 
(*x 

cally 

A*,y,p) = o. (2) 

308. There are three general methods which should be made use 
of in solving (1): 

(1). Solve for y ; (2). Solve for x ; (3). Solve for p. 

309. Equations Solvable fory. — If (2) can be solved for y, the 
equation becomes 

y = F(x,p). (1) 

Differentiate with respect to x. 

8F dF dp 

•*• P = Zx-+WTx ( 2) 

dp 
This equation (2) is of the first order in — . 

The elimination oi p between (1) and the solution of (2) furnishes 
the solution of (1). The elimination of p is frequently inconvenient 
or impracticable. When this is the case, the expression of x and^y in 
terms of the third variable/ is regarded as the solution. 

EXAMPLES. 

1. Solve/ + 2xy = x 2 +y 2 . (i) 

.-. y = x+ \/p. 
Differentiating, 

, J dp 

428 



Art. 309.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 429 
or dx = — — * — . / 2 \ 

. M - 1 

•'• * = £log y - +', 

/*+ ' 
, I _L <.2*-2«: 

or /* = ~—L (3) 

Eliminating /, we have for the solution 

k + e 2x 

2. Solve x — yp = ap 2 . (!) 

Differentiate y = , with respect to x, and put the result in the form 



dx 1 ap 



dp P(i-p') i-f 

Solving this linear equation, 

p 

x = • (c + a sin-ip). (2) 

4/1 -^ 

Substituting in (1), 

y — _ a p -f (<: + sin-*/). (3) 

Vi - P 1 

The values of x, y expressed in terms of the third variable/ in (2), (3) furnish 
the solution of (1). 

3. Clairaut's Equation. — The important equation, known as Clairaut's, 

y=px+f(P), (1) 

can be solved in this manner. 

Differentiate with respect to x. 

... P=t + x f x+np) % 

or, [*+/'(/)] %. = «. (2) 

The equation (2) is satisfied by either 

dp 
x + f\P) = 0, or f x = o. 

The solution of (1) is obtained by eliminating p between either of these equa- 
tions and (1). 

dp 

~- = o gives p = c, constant. 

Therefore one solution is 

y = cx+f(c), (3) 

which is the family of straight lines with parameter c. 

The second solution is the result of eliminating p between 

y =px+f { p), 



and o = x +f'{p). S ^ 

The second of these equations is the derivative of the first with respect to/; 
x and y being regarded as constants,/ as a variable parameter. This result is 



43° INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX. 

clearly the envelope of the family of straight lines representing the first solution (3). 
This envelope is called the singular solution of(l). 

Thus the general solution of Clairaut's equation (1) is effected by substituting an 
arbitrary constant for/ in the equation. The singular solution is the envelope of 
the family of straight lines representing the general solution. 

4. Lagrange's Equation. — To integrate 

y = xf{p) + i\p\ (1) 

Differentiating with respect to x and rearranging, 

dp^ AP)-P ^ AP)-P 
This is a linear equation in x and can be solved by § 305. II, Ex. I. 
Eliminating/ between (1) and the solution of (2), the solution of (1) is obtained. 
Otherwise x and v are obtained in terms of the third variable/. 

5. Solve y - (1 -\- p)x + /'-'• 

dx 

Differentiating, 1_ ^ — _ 2/. 

dp 

Solving this linear equation, 

x = 2(1 — p) -\- ce~P\ 

.-. y = 2 -/ 2 + (i +p)ce-P. 

6. Solve x 2 (y —px) = yp 2 . 



dv 



which is Clairaut's form. 

.-. v — cu -j- c 2 . Hence y 2 — ex 2 -\- c 1 . 

310. Equations Solvable for x. — When this is the case 

A*> y> p) - ° 

becomes 

* = F(y,PY (*) 

Differentiate with respect to y. 

i___bF bF dp . 

■'■ j-Jt + lp4>' 

This is of the first order in -f . The elimination of p between (1) 

dy 

and the integral of (2), or the expression of x and y in terms of/, 

furnishes the solution of (1). 











EXAMPLES. 












1. 


Solve 


x = 


= ;'+/ 2 - 






















I dp 

— = I 4- 2/ 4- » "r 

/ T * dy 


dy = - 


. 2 pr,ip . 
p - 1 










y = 


c — 


L/ 2 + 2/ + : 


2log(/ - 


I)], x= 


- c — 


[2/ + 


2l 


>g(i>- 


0]- 


2. 


X = J 


' ~f" log/ a . A/is. 


y = c - 


a log(/ - 


- 0. 


X = C 


+ 


a log- 


/ 




















/ 


— I 


3. 


Solve 


p 2 y 


4- ^px = y. 








Ans. 


J 2 


= 2CX 


+ <- 2 









Art. 313.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 431 

311. Equations Solvable for />. — The equation /'(.\\ 1, />) = o is 
a quadratic in />. 

If this can be solved in a suitable form for integration tor/, it 
becomes 

\P- <t>{*>y)\\P -*(x,y)\ = o. 

Each of the equations 

p = 0(-v, r) and p = ip(x, y) 

is of the first order and degree in ~, and their solutions are solu- 

dx 

tions of (1). 

Such solutions have already been discussed. 

EXAMPLES. 

1. Solve p 2 — (x -\- y)p -f- xy = o. 

(/ -*)(P -J>) = 

dy 

gives </y — x dx = o, and dx = o. 

. •• 2/ = jr 2 -)- c, and j == r^. 

2. p 2 — $P + 6 = °- Ans - y = 2x ~\- c, y = 3^ + <:. 

312. In particular, iff(x, v, p) — o does not contain x or does 
not contain y, corresponding simplifications of the above processes 
apply, see § 306. 

312. Equations Homogeneous in x and y. — When the equation 
f(x, y, p) — o is homogeneous in x and_y, it can be written 

(1). Solve, if possible, for p and proceed as in § 297, II. 
(2). Solve fory/x> Then the equation becomes 

y = x/(p). (2) 

Differentiate (2) with respect to x and rearrange. 

. dx = f\p) dp 
* p-(fP)' 

EXAMPLES. 

1. Solve xp 2 — 2yp -f- ax — °- dns. 2cy = c 2 x 2 -f- #• 

2. Solve j = yp 2 + 2/x. ^4«j. y 2 = 20: -j- ^ 2 - 

3. x 2 p 2 — 2xj'^> — ly 2 = o. ^//j\ ry = x 3 , xy = c. 

Orthogonal Trajectories. 

313. A curve which cuts a family of curves at a constant angle is 
called a trajectory of the family. We shall be concerned here only 
with orthogonal trajectories. If each member of a family of curves 



43 2 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX. 

cuts each member of a second family of curves at right angles, then 
each family is said to be the orthogonal trajectories of the other family. 
At any point x, y where two curves cross at right angles, the rela- 
tion pp' = — i exists between their slopes/, p' . 

314. To Find the Orthogonal Trajectories of a given Family of 
Curves. 

Let <P( x t y> a ) = o ( 1 ) 

be the equation of a family of curves having for arbitrary parameter a. 
Let f(x,y,p) = o (2) 

be the differential equation of the family (1), obtained by the elimina- 
tion of the parameter a. 



The differential equation 



f(*,y, -))=°, 
or . f[f>y> - -§) =0 > < 3) 

is the differential equation of a family of curves, each member of 
which cuts each member of (i)at right angles. Therefore the general 
integral of (3), 

ip(x,y, b) = o, (4) 

is the equation of the family of orthogonal trajectories of (1). 

EXAMPLES. 

1. Find the orthogonal trajectories of the family of parabolze y 2 — /\.ax. 
Differentiating and eliminating a, the differential equation of the family is 

dy _ y 
dx 2x 
The differential equation of the orthogonal trajectories is 

dx y 
dy ~ 2x' 
The integral of which is x 2 -f- \y % = c 2 , a family of ellipses. 

2. Find the orthogonal trajectories of the hyperbolae xy = a 2 . 

The differential equation is y + xp — o. The differential equation of the 
orthogonal trajectories is 

dx 
y -' Ty= °' 

giving the hyperbolae x 2 — y 2 — c 2 for trajectories. 

3. Find the orthogonal trajectories of y = mx. 

4. Show that x 2 -\- y 2 — 2cy =0 is orthogonal to the family 






Art. 316 ] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 433 

5. Find the orthogonal system of — +<- — 1, in which 6 is the parameter. 

Aus. x- -f /-' = a 2 log x 2 +c. 

6. Find the system of curves cutting x 2 -f Py* — b 2 a~ at right angles, a 
being the parameter of the family. Ans. yc = x^. 

The Singular Solution. 

314. We have seen in the rase of Clairaut's equation, § 309, Ex. 3, 
that there may exist a solution of a differential equation which is not 
included in the general solution. Such a solution, called the singular 
solution, we now propose to notice more generally. 

315. Singular Solution from the General Solution. 

Let <t>(x,y, c) = o (1) 

be the general solution of the differential equation 

f(x,y,p) = o. (2) 

A solution of the differential equation (2) has been denned to be 

an equation (1) in x, y such that at any point x, y satisfying the 

dy 
equation (1) the x,y, and p = —derived from this relation satisfies (2). 

The general solution (1) being the integral of (2) satisfies the con- 
dition for a solution. Also, however, the envelope of the system of 
curves (1) is a curve such that at any point on it the x, y, p of the 
envelope is the same as the x,y, p of a point on some one of the sys- 
tem of curves (1), and must therefore satisfy (2). Consequently the 
envelope of the family (1) is a solution of (2). 

This is a singular solution. It is not included in the general 
solution, and cannot be derived from it by assigning a particular 
value to the parameter c. 

We may then find the singular solution of a differential equation 
(2) by finding the envelope of the family (1) representing the general 
solution of (2). 

Thus the singular solution of (2) is contained in 

tp(x,y) ■= o, 
which results from the elimination of c between 

(p(x, y, c) = o and <p[{x, y, c) = o. 

316. Singular Solution Directly from the Differential Equa- 
tion. — It is not necessary to obtain the general solution of a differen- 
tial equation in order to get the singular solution. The singular solu- 
tion can be obtained directly from the differential equation without 
any knowledge of the general solution. 

Let the differential equation 

f(x,y, p) = o (1) 



434 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX. 

be regarded as a family of curves having the variable parameter/*. 
Find the envelope 

X (x,y) = o (2) 

of (1), as the result of eliminating p between 

A x > y>P) = ° and fi( x > y> p) = °- 

dv 
Since at any x,y satisfying (2) the x,y, — of (2) is the same as 

dy 
the x,y, -—-of a point on (1), the equation (2) must contain a solu- 
dx 

tion of (1). 

EXAMPLES. 

1. Find the general and singular solutions of p 2 -j- xp = y. 

This is Clairaut's form, and the general solution can be written immediately by- 
putting/ = const. 

However, independently, we have on differentiation 

dp 

dp 

-v- = O gives p — c, and y = ex -j- c 2 for the general solution. Differentiating 

with respect to c and eliminating c, we find the singular solution 4)' -)- x 2 = o. 

Integrating the other factor, x -\- ip = o, or eliminating/ between this and the 
differential equation, the same singular solution is found. 

2. Find the general and singular solutions of the equation y = px -j- a 4/1 -\~ p 2 . 

Am. x 2 -f- y 2 = a 2 . 

3. Find the singular solution of x 2 p 2 — 2> X )'P + 2 J>' 2 -)- ^r 3 — o. 

^;zj. jr 2 (j' 2 - 4T 3 ) == o. 

317. The Discriminant Equation. — The discriminant of a func- 
tion F[x) is the simplest equation between the coefficients or constants 
in i^) which expresses the condition that /"has a double root. If 
F has two equal roots, equal to a, then 

F(.v) = (x - ay<t>(x), 

where is some function which does not vanish when x = a. Hence, 
differentiating and putting x = a, we have the conditions for a double 
root at a, 

F(a) = o, F'(a) = o, F'\a) ^ o. 

Eliminating a between /iV) = o, F\a) = o. or, what is the same 
thing, eliminating x between F(x) = o, F'(x) = o, we obtain the 
discriminant relation between the coefficients, the condition that 
F(x) shall have a double root. 

318. ^-discriminant and /-discriminant. 

Let cp(x, y, c) = o be the general solution of the differential 
equation y(jt, y, p) = o. 



Art. 320.J EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 435 



(1). The equation t/)(x, y) = o which results from the elimination 
of c between the equations 

(p(x, y, c) — o and (fi' c (x, y, c) = o 

is called the c-discriminant, and expresses the condition that the equa- 
tion <fi = o, in c, shall have equal roots. 

(2). The equation x{ x >y) = ° which results from the elimination 
of/ between the equations 

f(x, y, p) = o and /;(x, y, p) = o 

is called the /-discriminant. It expresses the condition that the equa- 
tion/" = o, in/, shall have equal roots. 

319. c-discriminant contains Envelope, Node-locus, Cusp- 
locus. — The c-discriminant is the locus of the ultimate intersections 
of consecutive curves of the family cf)(x, y, c) = o. 

It has been previously shown that the envelope of the family is 
part of this locus, and also that the envelope is tangent to each member 
of the family. 

Suppose the curves of the family have a double point, node, or 
cusp. Then, in case of a node, two neighboring curves of the family 




Fig. 155. 

intersect in two points in the neighborhood of the node, which con- 
verge to the node-locus as the curves converge together. In the neigh- 
borhood of the envelope two neighboring curves intersect in general 
in but one point. 

In the case of a cusp, two neighboring curves intersect, in general, 
in three points in the neighborhood of the cusp-locus. Two of these 
points may be imaginary. 

We may expect to find the envelope occurring once, the node-locus 
twice, the cusp-locus three times as factors in the c-discriminant 

320. /-discriminant contains Envelope, Cusp-Locus, Tac-Locus. 

■ — If the curve family f(x,y, p) ~ o has a cusp, then for points along 
the cusp-locus the equation vanishes for two equal values of/, as it 

does also for points along the envelope. But, in general, the — of the 



cusp-locus is not the same as the/ of the curve family and therefore 
does not satisfy the differential equation. 



43 6 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX. 

Again, at a point at which non-consecutive members of the curve 
family = o are tangent the x, y, p of the point satisfies the equa- 
tion f = o. The locus of such points is called the lac-locus. The 

dy 
— of the tac-locus is not the same as that of the curve family = o, 

and the tac-locus therefore is not a solution of/" = o. 

321. It has been shown by Professor Hill (Proc. Lond. Math. Soc, 
Vol. XIX, pp. 561) that, in general, the 

f the envelope once, 
c-discriminant contains -J the node-locus tw ice, 

(the cusp-locus three times, 

C the envelope once, 
/-discriminant contains 1 the cusp-locus once, 
(the tac-locus twice, 

as a factor. This serves to distinguish these loci. Of these, in 
general, the envelope alone is a solution of the differential equation. 
It may be that the node- or cusp-locus coincides with the envelope, 
and thus appears as a singular solution.* The subject is altogether 
too abstruse for analytical treatment here. 




EXAMPLES. 

1. xp 2 — (x — a) 2 = o has the general solution 

y + c = i-*- 3 - 2ax K 
90' + c ? = 4*(* - 2 a Y- 
The /-discriminant condition is x(x — a) 2 = o, the 
^-discriminant condition is x(x — 3a) 2 =0. x = o occurs 
once in each, it also satisfies the differential equation and 
is the singular solution or envelope, x = a occurs twice 
in the /-discriminant and does not occur in the c dis- 



therefore the tac-locus. x = 3^ 
the c- and does not occur in the/dis- 
: T> a is therefore a node locus. 
Show that [y -j- c) 2 = x 3 is the general solution of 
o is a cusp-locus. 



There 



cnminant. x = 
occurs twice in 
criminant. x = 

2. 

\p 2 =: gx, and x 3 = 
singular solution. 

3. Solve and investigate the discriminants in 
p 2 + 2xp = y. 

General solution (2x* -]- 3x1' -f- c) 2 = 4(x 2 -(- j) s . No 
singular solution. Cusp-locus x 2 -\-y =0. 

4. In %ap s = 27J', show that the general solution is 
ay 2 = (x — cf, singular solution y — o, cusp-locus j 3 = o. 

5. Find the general and singular solution of y = xp — p 2 . 

Ans. y = ex — c 2 , x 2 = 41'. 



Fig. 156. 



*Proc. Lond. Math. Soc, Vol. XXII. p. 
node- and cusp-loci which arc also envelopes.''' 



216. Prof. M. J. M. Hill, 



On 



Art. 321. J EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 437 



EXERCISES. 

Find the general solutions of the following equations. 

1. p 2 = ax 3 . Ans. %${y 4 cf = 4^. 

2. p 3 = <wr*. ^«j. 343(j 4 cf = 270x1. 

3. p'\x + 2y) 4- 3/ 2 (x +^) +/(/ + 2.r) = o. Factor and solve. 

Ans. y = e, x + y = c, xy -f x 2 4 / 2 = f. 

4. / 2 — 7/ 4 12 = o. ^4«j. y = 4JC -\- c. y = 3Jf + r. 

5. ^ 2 — 2// -f- ax = o. ^«j-. 29' = At 3 -j- rt. 

6. yp 2 + 2x/ = j. Ans. y 2 = icx 4 r 2 . 

7. x 2 / 2 — 2jrj// 4 2y' 1 — x 1 = O. Ans. sin-i — = log ex. 

x 

8. y =p(x- b) +j. Ans. y = c{x - b) -f-i 

9. xv 2 (/ 2 + 2) = 2// 3 + x 3 . Ans. (x* - y 2 + r)(* 2 - j 2 4 ^) = o. 
10. y 4 /jt = .r 4 / 2 . ^4«j. xy = <r -f- c 2 x. 

11. ayp 2 -\- (2x — b)p — y = O. Ans. ae 2 4 e(2x — b) — y 2 = o. 



12. / — /x = 4/1 +/ 2 /(-* 2 -f-.y 2 )- Change to polar coordinates. 

Ans.B + c = f MVP . 

13. (*? - j) 2 = «(i 4/ 2 )(* 2 +y) s - 

^«j. tan -1 (- <: = vers -1 2a f/.r 2 -(- / 2 . 

14. {xp — y) 2 = p 2 — 2 —p 4 I. Ans. sin— * — = sec-'x 4 <r. 

15. 3/ 2 j 2 — 2x>'/ + 4/ 2 - x 2 = o. Put .r a — 3/ 2 = z^ 2 . 

Ans. 3(x 2 -j- j 2 ) ± \cx 4 r 2 = o. 

16. (x 2 + j 2 )(i +/) 2 - 2(x + y)(i + /)(* +#) + (x +ypf = o. 

^«j. x 2 +y 2 — 24* 4 y) 4 ^ 2 = o. 

17. x H — = a. ^«f. (y + f, 2 4- (* - a) 2 = 1. 

^/i+/ 2 

18. y = px -\-p — / 3 . Ans. y = ex 4 c — r 3 . 

19. j 2 - 2/A7 — 1 = p 2 (i — x 2 ). Ans. (y - ex) 2 = 1 -\- c 2 . 

20. y = 2px 4 y 2 p 3 . Put y 2 = z. Ans. y z = ex -\- ^c 3 . 

21. x 2 (y - px) = yp 2 . Ans. y 2 = ex 2 4 c 2 . 

eh 2 

22. {px -y){py 4 *) = /fc 2 /. ^»j. / 2 - or 2 = - — -. 



23. y = xp 4 -t/£ 2 4 a 2 / 2 . ^»j. v = ex 4- |/^ 2 + « 2 ' 2 i 

singular solution x 2 /« 2 -\-y 2 /b 2 = 1. 

24. ^ = p(x — b) 4 a//, singular solution, j 2 = 4a(x — b). 

25. (^ — xp)(mp — n) — mnp. Ans. (y — ex)(me — n) = nine, 

singular solution, (x/m)^ ± {y/nf- = 1. 

26. y 2 - 2xyp + (1 + x 2 )p 2 = I. Ans. (y - ex) 2 = 1 - e 2 , 

singular solution, y 2 — x 2 = 1. 





Ans. 


y 




CX' 


t" 




X 2 


- 


} ,2 
C 2 


= 


I. 


a? 


- c 2 



438 INTEGRATION FOR MORE THAN ONE VARIABLE. {Ch. XXXIX. 

27. / 3 - 4-r;'/ -f 8i< 2 = o. Ans. y = c{x - c)\ 

singular solution, 2jy = 4^. 

28. Find the orthogonal trajectories, A being the variable parameter, of the 
following curve families: 

(1). ~ f jz = 1. Ans. x 2 + y 2 = a 2 log x* 4- c. 

(2). * 2 + ni l y 2 = m*\\ 

29. Find the orthogonal trajectories of the circles which pass through two fixed 
points. Ans. A system of circles. 

30. Find the orthogonal trajectories of the parabolse of the «th degree 

a n-iy — x n . Ans. ny' 1 -f- x 2 = c 2 . 

31. Find the orthogonal trajectories of the confocal and coaxial parabolae 

y 2 = \\{x -j- A). Ans. Self-orthogonal. 

32. Find the ortho-trajectories of the ellipses x 2 /a 2 -f- y 2 /b 2 = A 2 . 

Ans. y b% = ex"*. 

33. Show that if 

is the differential equation of the family of polar curves cp(p, 6, c) = o, then 



(aw|)=o 



is the differential equation of the orthogonal system. 

34. Find the orthogonal trajectories of/? = a(i — cos 0). 

Ans. p = c(i -\- cos 6). 

35. Also the ortho-trajectories of — 

(1). p n sin nQ = a n . Ans. p n cos nB = c H . 

(2). p =z log tan 6 -f- a. Ans. 2/p = sin 2 6 -{- c. 



CHAPTER XL. 

EXAMPLES OF EQUATIONS OF THE SECOND ORDER AND 
FIRST DEGREE. 

322. The differential equation of the second order and first degree 
is an equation in x y y, p, q, 

A x > y> P> 9) = °> 

dy dp d 2 y 

where p = — , q = — = — 2 , and in the equation q occurs only 

in the first degree. 

We shall attempt the solution of the equation for only a few of the 
simplest cases. 

We have seen that the general solution of the equation of the first 
order and degree gave rise to a singly infinite number of solutions, 
represented by a family of curves having a single arbitrary parameter, 
this parameter being the constant of integration. 

In like manner, the general solution of the equation of the second 
order and first degree, involving two successive integrations, requires at 
each integration the introduction of an arbitrary constant. The 
general solution, therefore, contains two arbitrary parameters, and is 
correspondingly represented by a doubly infinite system of curves, or 
two families, each having its variable parameter. 

The process by which a differential equation of the second order is 
derived from its primitive is as follows. 

Let <P(*,y, c lf c 2 ) = o (1) 

be an equation in x, y and two arbitrary constants c v c 2 . Differ- 
entiating (1) twice with respect to x, there results 

av d y dy ay /^_y a/ ^ _ 



dx 2 dx dy dx dx 2 \dx J dy dx 2 

Between these three equations can be eliminated the two arbitrary 
parameters c v c 2 . The result is the differential equation of the 
second order, 

A x > y< a 9) = o. 

439 



44° INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XL. 

EXAMPLE. 

The simplest equation of the second order is 

d * y - h 
Here the integrations are immediately effected. 

<■(£)-** 

... £_*,+ * 

c l being the first constant of integration. Integrating again, the general solution is 

y = %&x 2 -f- c x x -f- <V 
The two arbitrary parameters c lf c 2 giving a doubly infinite system of parabolae. 

323. The Five Degenerate Forms. — The ordinary processes of 
integrating differential equations are of tentative character. We are 
led to the solution of general forms through the consideration of the 
simpler cases. Investigation of the general methods of treating this 
subject is out of place in this text, and we shall consider here only a 
few interesting and important equations of simple form. 

A general method of solution can be proposed for the five degen- 
erate forms of the general equation, 

1. f(x, q) =: o; 2. f(y, q) = o; 3. f(p, q) = o; 

4. f(x, p, q) = o; 5. f[y, p, q) = o. 

324. Form/"(.r, q) = o. — This being of the first degree in q, 

dx> = F ^- 

The differentials involved are exact, and it is only a question of 
integrating twice. The solution is 

. •. y = / dx I F(x)dx -\- c x x -f- c r 
Ex. q = xe x . Ans. y = (x — 2)e* -\- c x x -f- c y 

325. Form/(7, q) — o. — Here 

PutfU/. Then £=*=**=,* 
dx dx z dx dx dy dy 

The equation becomes 

pdp = F{y)dy. 



Art. 326.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 441 
Integrating, 

dy 



dx = 



The integral of this gives the solution. 



1. Solve -— = a 2 y. 
dx 2 



EXAMPLES. 



2 / I\y)dx = a 2 y 2 . Put c x — a 2 c. 
dy 



.-. dx = 

a ^y 2 + c 

Hence ax = log(y + ^y 2 -\- c) -f- r 2 . 

Show that this can be transformed into 

y = c{e ax -f cj<r-"x. 

dy 
Multiply the given differential equation by 2 -^-. 



1%=tM' ~4.-*i<» 



Hence the firs{ integral is, as before, 
2. Solve ^ 2 + a2 y = o. 



Here 


2 \ J\y)dy = - a 2 y 2 . Put *, = a 




■ adx- dy 




\/c 2 - y 2 


Hence 


y 
ax 4- c <> = sin -1 — , 
2 c 


or 


y = c sin (ax 4- ^ 2 )> 



Multiply the differential equation by 2p and obtain the first integral directly as 
in Ex. 1. 

Examples 1 and 2 are important in Mechanics. 

3. Solve q \/ay = I. Ans. 3* = 2a\y^ — 2c 1 )(y* -{- cj* -\- c Y 



326. Form/(/>, q) = o. 

), or 



-«" %=*(£)■ « l=^>' 



... „/ 



•n/) ' 



44 2 INTEGRATION FOR MORE THAN ONE VARIABLE.. [Ch. XL. 

This is an equation of the first order, the solution of which is that 
of the required equation. 

EXAMPLES. 



It 
dy 



1. Solve 

dx 1 

Integrating p—*dp -\- adx, we have for the first integral 

dx 



ax -J- c 
.-. y = log (ax + c) + d, 
or ey = c x x -+- c % . 

d*y dy — 

2. Solve a -±- = -f. Ans. y = c x e a -f- c 2 , 

dx 2 dx 

3. q = p' 1 + I. Ans. e~y = c 2 cos (x -f- f i). 

4. <7 + / 2 + i = o. Ans. y — log cos (x — c x ) -\- r 2 . 

327. Form f(x, p, q) = o. — Such equations are reduced to the 
first order in x and/ by the substitution q = -J-. 



"- A*>P>!) =/(^A ^) = 



EXAMPLES. 

d*y dy 

^ m (I -\- x 2 ) -^- 2 -{- x •— -\- ax = o is equivalent to 



± 4- x p 4- -^- 
The first integral is 



The second integration gives 

^ = * 2 — <z* + ^ log (* + |/i + * 2 ). 

2. (1 + * 2 )? +/ 2 + 1=0. ^«j. y = c x x+{cf + 1) log {x - + *r 
328. Form f(y, p, q) = o. 

_d 2 y _ dp dy dp _ dp 
~ dx 2 ~ dx ~~ dx dy — dy' 

Substituting for q, the equation is reduced to the first order my 
and/. 

EXAMPLES. 

l.^+/l = o. Ans. 1±Z = *«(*+*). 

ax 2 dx a — y 

*' £ + (£)= *■ *». •-* + «*. + * 

3. yq — P' 2 = ^ 2 log 7. ^«j. log J = c x e* -)- <" 2 <f-*. 



Art. 329.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 443 
329. Solution of the Linear Equation 

in which A, B are constants. 

The solution of this equation is suggested by the solution of the 
corresponding equation of the first order 

dy 

V- ay = o, 

dx 

which gives — = — adx, the solution of which is y = ce~ ax . 

If we try y = e mx in (1), we have 
<Pe mx de mx 

~dx^ + A ~dx~ + BemX ~ ^ + Am + *>«"**■ (2) 

I. ^tftfAf of the Auxiliary Equation Real and Unequal. — The func- 
tion (2) vanishes if ?n be one of the roots of the auxiliary equation 

m 2 -f- Am -j- B = (m — m x )(m — m„) = o. (3) 

Hence y = e^i^is a solution. Also, jy = ^'"1* is a solution for 

any arbitrary constant c v In like manner y = c 2 e m * x is a solution. 
The sum of these two, 

^ = ^i* -f- c 2 e M **, (4) 

is also a solution, and is the general solution of (1) since it contains 
two independent arbitrary constants, c x and c r 

II. Roots of the Auxiliary Equation Real and Equal. — If m l = m 2 , 
the solution (4) fails to give the general solution, since then 

and c x -f- c 3 = c' is only one arbitrary parameter. 

The solution in this case is immediately discovered on differentiat- 
ing (2) with respect to m. For then 

d 2 xe" tx dxe 



mx 



-f A -f Bxe mx = (2m -f- ^Jg"^ -f (w 2 -|- Am + B)xe mx . 

It m = /a is the double root of (3), then (3) and its derivative 
vanish when m = ju. Consequently y = .are' 1 -* is a solution, and 
also is y = £#£'*•*. Hence the sum of the two solutions cV*and £*#"' 
is the general solution of (1) when /a. is a double root of (3), or 

,, = <^( c ' _|_ cx). (5) 

III. 7?oo/j- of the Auxiliary Equation Imaginary. — When the roots 
of (3) are imaginary and of the forms 

m l = a-\- id, m 2 = a — id, 



444 INTEGRATION FOR MORE THAN ONE VARIABLE. . [Ch. XL. 

where i = \/ — i, these roots may be used to find the solution. For 
(4) becomes 

y — Cl e {a + ib ^ + c/ a ~ ib)x , 
= e ax {c x e ibx -f c 2 e~ ibx ). 

We have by Demoivre's formula 

e ibx — cos j) X _|_ i s j n i x ^ 

e -ibx __ cos fa _ 1 s [ n l Xt 

Therefore the solution is 

y = e ax { (c 1 + c 2 ) cos bx -f- (^ — c 2 ) z' sin &r}, 

= e ax (k l cos for -f- k 2 sin for), (6) 

where ^ = ^ -f- c 2 , k 2 = (c l — c 2 )i. If the arbitrary constants 
c x and c 2 be assumed conjugate imaginaries, the constants k and k 2 are 
real. 

By writing tan a = £j/£ 2 , or cot/? = ^/^ 2 , the solution (6) 
may be written respectively 

y — c'e ax sin (for -f- a), 

= c"*"* cos(for - /?). (7) 

EXAMPLES. 

1. Solve q — p = 2y. 

The auxiliary equation is 

m 2 — m — 2 = (m -j- l)(^ — 2) = O. 
The general solution is therefore jr = c x e~ x -j- ^ 2 *. 

2. If ? - 2/ 4-/ = 0, (m— I) 2 = o, .-. y = e*(c x + ^). 

3. Solve q -\-2>p = SW- 

m i _j_ yn — 54 = (m — 6)(m -f 9). 
.-. y = c l e 6x -j- c 2 e— 9 X . 

4. Solve q -j- 8/ -f 257 = O. 

/w 2 -f- 8/« -f- 25 = o gives m = — 4 ± 3 |/— 1. 
. •. j — ^— 4^(^ cos 3 jt -4" <£ 2 sin 3^). 

330. Solution of the Equation 

y4, i? being constants. 

Put jc — £ z , then z = log #. Also, 

r/r ^dfe _ 1 </>' </^ 1 /VV ^\ 
dx dz dx x dz ' dx 2 x 2 \dz 2 dz) ' 
On substitution, equation (1) becomes 

which is the form solved in § 329. 



Art. 331.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 445 

Ex. Solve x 2 q — xp -{-y = o. 
The equation transforms into 

d*y dy , 

331. Observations on the Solution of Differential Equations. 

— The remarks made on the integration of functions are equally 
applicable to the integration of differential equations. The process 
is of tentative character, and skill in solving equations comes through 
experience and familiarity with the known methods of solving the 
integrable forms. 

When the equation is not readily recognizable as one of the stand- 
ard forms for solution, it can frequently oe transformed into a recog- 
nizable form by substitution of a new variable. 

Most of the processes given in this chapter for the solution of 
certain forms of the equation of the second order and first degree are 
immediately applicable to equations of higher orders. In the exer- 
cises will be found certain simple equations of higher order than the 
second, proposed for solution by the methods exposed in the text. 

General methods of solving differential equations must be reserved 
for monographs on the Theory of Differential Equations. 



EXERCISES. 



d 2 y 



1. -^ - a 2 x -f b 2 y. Put a 2 x -f b 2 y = z, etc. 

Ans. a 2 x -f b 2 y = c^* -f- c 2 e~ bx . 

d 2 v 

2. — — = a 2 x — b 2 y. Ans. a 2 x — b 2 y = c. sin bx 4- ^9 cos bx. 

dx 2 * 

3. q = ev. 



\/2ey 4- c 2 — c x \y _ V 2 

— > e — 

y2ev -f- c 2 -j- c x H — ■ 

or 2ef = c 2 sec 2 ^* -f- c 2 ), according as the first constant of integration is 

+ c 2 , o, or -c 2 . 

4. xq -\-p = O. Ans. y = c x log x -f c r 

e % dx -j- c r 

6. x 2 q — 2y. Put z = 2y/x 2 . .'. xy = qx 3 -f- c v 

7. q + 127 = Jp. Ans. y = c x es* + c 2 e**. 

8. 3(q + y) ' — 10/. Ans. y = c x e& + c^ x . 

9. q + \p - y. Ans. ye** =C x e* VI + c 2 e-x Vs . 

ax bx 

10. ab(y -f q) = (a 2 -f b 2 )p. Ans. y = c x e b + c 2 e*. 

11. ^L = if. Ans. y = c x &* + c+-** + c y 

dx s dx 



446 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cir. XL. 

12. q — 6p -J- i$y = o. Ans. y — [c x sin 2x -\- c 2 cos 2xyv c . 

13. q — lap -)- b 2 y = o. Ans. According as a > or < b, 

y — cax(c x gxV a *-i>* _|_ c 2 e- x ^*' 1 -P i ), or ^(q sin .r \/b' 2 — a 2 + <r 2 cos.r |/£ 2 — a 2 ). 

14. ? - \abp -f- (rt 2 + 3 2 ) 2 / = o. 

Ans. y = e 2abx \c l sin {a 2 — 3 2 ).r -|- c 2 cos(a 2 — ^ 2 )x|. 

15. q — P log a 2 -\- [1 -J- (log #) 2 ] j =r o. ^«j. y = tf*^ sin x -\- c 2 cos jc). 

16. </ — 2a/ -(- <z 2 j = o. ^«j . y = ^a^fq -f c^x). 

17. q — o. Ans. y z=. c x -\- c 2 x. 

18. —J = 4!?. Ans. y = c x e±* + r 2 + f 3 *. 

19. # 2 ^ — xp = $y. ^4»j. xj' = ^j-r 4 -|- r 2 . 

20. {a + for) 2 ? + ^(« + bx)p + ^ = O. 

Ans. y — c x sin log (a -f- £;r) -f- <: 2 cos log (a + t> x )- 
cPy 



21. .r -7-3- = 2. ^4«j. 7 = c x -f- f 2 jf -)- c z x 2 -|- x 2 log 



</jr> 



.r. 



22. -^ — sin 3 .r. A;is. y = c x -\- c 2 x -j- c % x 2 -f- |cos .r — ^cos 3 x. 

23. £V 3 = a. Ans. (c x x + c,) 2 = ^j 2 - a. 

JC X 

24. a 2 ^ 2 = 1 -f- P"' Ans. 2y/a = c x e<* -(- ^-'^ rt -j- r 2 . 

25. «V 2 = ( J +/ 2 ) 3 . >*»*• (•* + <i) 2 + ( y + ' 2 ) 2 = " 2 - 

26. (1 — x 2 )q — xp = 2. Ans. y = c x sin— j jt -|- (sin- x Jf) 2 -f- ^ 2 - 

27. ^/+/ 2 =i. ^«j. y 2 = x 2 + c x x + c 2 . 

28. (1 - log j)^ -f (1 + \ogy)p 2 = o. Ans. [c x x + r 2 )(log y - 1) = 1. 

29. ^'<7 — / 2 =/ 2 log;/. Ans. log/ = q** -(- c 2 e~ x . 



30. (/ - ^/) 2 = 1 + q 2 . Ans. y - U x x 2 + x 4/1 + r x 2 -f r 2 . 

31. Find the curve in which the normal is equal and opposite to the radius of 
curvature. [Catenary.] 

32. Find the curve in which the normal is equal to the radius of curvature and 
in the same direction. 

33. Find the curve in which the radius of curvature is twice the normal and 

opposite to it. The parabola, x 2 = 4i(y — c). 

34. Determine the curve in which the normal is one half the radius of curva- 
ture, and in the same direction. 



c- y 



The cycloid x -f- c sin— 1 -f- 4/2 o' — y' 1 = o. 

35. Find the locus of the focus of the parabola y 2 ■= ^ax as the parabola rolls 
on a straight line. [Catenary.] 

36. Find the locus of a point on a circle as it rolls on a straight line. 

37. Express the locus of the center of an ellipse as it rolls on a straight line in 
terms of an elliptic integral. 

38. The problem of curves of pursuit was first presented in the form : To find 
ith described by a dog which runs to overtake its master. 



Art.33I-] e Q uations OF SECOND ORDER AND FIRST DEGREE. 447 

The point A describes a straight line with uniform velocity; it is required to find 
the curve described by the point B, the motion of which is always directed toward 
A and the velocity uniform. 

Take the path of A for j-axis. The tangent intercept on the /-axis is y — xp. 
By hypothesis the change of this is proportional to the change of arc-length. 



.-. — x dp = m f/i + p' 1 dx, 

log ** + log (/ + \/l 4- /-') 4- log € x = O, 





2/ 


= 


c r 


\ x ~m 


— 


V", 






"•2 ~ 


C l 


X 

m 


11+ 1 


- 


c r 


x~ 


-W+I 




»l 


— I 



The curve is algebraic, except when m = 1, then we have to substitute log x 
for — x-™+*/(m — I). 



APPENDIX. 
SUPPLEMENTARY NOTES. 



449 



APPENDIX. 

NOTE 1. 

Supplementing § 30. 

Weierstrass's Example of a Continuous Function which has 
nowhere a Determinate Derivative.* 

The function 

00 

J\x) = ^ b n COS (a n 7TA-) } 

o 

in which x is real, a an odd positive integer, b a positive constant less 
than 1 , is a continuous function which has for no value of x a deter- 
minate derivative, if ab > 1 -f- f 7T. 

Whatever assigned value x may have, we can always assign an 
integer j.i corresponding to an arbitrarily chosen integer ??i, for which 

- i < a m x - /* ^ + -J, 

Put jf ;w+i = tf' w Jtr — yu, and let 

... x >- x= _l±3n±l } x "-x-= 1 ~ Xm+l , 

a" 1 a m ' 

and .#' < .r < #". 

The integer m can be chosen so great that x' and x" shall differ 
from .v by as small a number as we choose. 

We have 

00 

f{x') — J\x) V~^ cos {a n 7tx') — cos (tf M 7r.r) 



;// — 1 



(ai)' 



COS (i7 M ^'^ / ) — COS (a n 7tx) 
a n {x' — x) 

cos {a m+n 7rx') - cos (« W+M 7r^) 



\ bm+n COS jfl 7TX J - COS (fl- ■ -g^ 



n=o 



* Taken from Harkness and Morley, Theory of Functions. 

45 



45 2 APPENDIX. 

Since 

cos ( a" 7t x) — cos (a n 7tx) . x -\- x \ \ 2 I 

a\x' - x) \ 2 I x' - x 

2 

and since^the absolute value of the last factor on the right is less than 
1, then the absolute value of the first part of (i) is less than 

in — 1 

o 

and therefore less than — = — — , if ab > 1. 

ab — 1 

Also, since a is an odd integer, 

cos (a m+n 7tx') — cos [a M (ju — i)n~\ — — ( — i)* 4 , 
cos (a w+n 7tx) = cos (a"jj7r -j- ^Wi^) = ( — i) M cos («"*,+,»). 
Therefore 



CO 

E 



cos (fl'-'^o') — COS (a'" +w ^Jt') 



.%- — j; 



= ( _ i^oty-V* ;+^Kv. t ), 



All the terms under the 2 on the right are positive, and the first 
is not less than §, since cos (x m+1 7t) is not negative and 1 -j- x m+J 



lies between 4- and j. 



Consequently 

/(*') -/(X) _ . ., „ m z(2 , 7T1J 



(- D-H-* g+a^), (n) 



Jt" — „V 

where £ is an absolute number > 1, and 1} lies between — 1 and -f- 1. 
In like manner 

where £,' is a positive number > 1, and rf' lies between — 1 and -j- 1. 
U ab be so chosen as to make 

ab > 1 + -|tt, 

2 7T 

that is, - > -7 , 

3 ab - 1 

the two difference-quotients have always opposite signs, and both are 
infinitely great when m increases without limit. Hence _/"(.\) has 
neither a determinite finite nor determinate infinite derivative. 



SUPPLEMENTARY NOTES. 



453 




Every point on such a line, if line it could be called, is a singular point. 
Some idea of the character of the geometrical assemblage of points representing 

such a function can be obtained by selecting 
two particular fixed points A, B of the as- 
semblage. Between A and />, in progressive 
order, select points P v P.,. . . . representing 
the function corresponding to x v j„, . . . 
Consider the polygonal line AI\ P., . . . B. 
Increase the number of interpolated points in- 
definitely, and at the same time let the dif- 
ference between each consecutive pair con- 
verge to o. Then, since the function J\x) 
is continuous, each side, P,P, „ of the broken 
line converges to o. But, instead of each 
angle between consecutive pairs of sides of 
this polygonal line converging to two right 
angles, it, as their lengths diminish indefi- 
nitely, as was the case when we defined a curve with definite direction at each point; 
let now these angles converge alternately to o and in. The polygonal line folds 
up in a zigzag. The point P converging to the neighborhood of a true curve AB. 
But the difference-quotient at any. point of the zigzag assemblage has no limit, it 
becomes wholly indeterminate as the two values of the variable converge together. 
It is also possible that the length representing the sum of the sides of the polygonal 
between any two points of the assemblage at a finite distance apart (however small) 
is infinite in the limit. 

Such functions are but little understood and have been but little studied. It is 
possible that they may have in the future far-reaching importance in the study of 
molecular physics, wherein it becomes necessary to study vibrations of great velocity 
and small oscillation. 

NOTE 2. 

Supplementary to § 42. 

Geometrical Picture of a Function of a Function. 



00*; 



can represent the function z 



If z =f(y), where y 
geometrically as follows: 

Draw through any fixed point in space three straight lines Ox. 
Oy, Oz mutually at right angles, so that 
Ox, Oy are horizontal and Oz is vertical. 
These lines fix three planes at right 
angles to each other. xOy is horizontal, 
xOz and yOz are vertical. 

The relation^ = <p(x) can be repre- 
sented by a curve P'Q' in the plane 
xOy. At any point P' on this curve we 
can represent z by drawing P'P =f(y), 
up \i/(y) is positive, down if J\y) is 
negative. The relation z —/(y) is 
represented by the curve P'"Q'" inyOz. 

as a function of x, is represented by the 

curve P" Q" in xOz. In other words, z as a function of x and y is 




Fig. 158. 



454 



APPENDIX. 



represented by a point in space having the corresponding values 
z, y, x as coordinates with respect to the three planes. The assem- 
blage of points representing z, y, x is a space curve PQ. The 
orthogonal projections on the three coordinate planes of PQ represent 
the functional relations 

(P'Q'), y=<P(x); (P"Q"), z=/\ ( t>(x)\; (P'"Q'"), *=Aj>)- 

The derivative D x y is represented by the slope of P'Q' at P' to 
Ox. The derivative D y z is represented by the slope of the tangent 
to P'"Q'" at P'" to Oy; the derivative D x z by the slope to the 
axis Ox of the tangent at P" to P"Q" . 

The function of a function is represented by a curve in space. 



NOTE 3. 

Supplementary to § 56. 

The nth Derivative of the Quotient of Two Functions. 



Let y = u/v. Then 
this product, we have 



vy. Applying Leibnitz's formula to 



u = vv 



u' 


= 


I 


? 


-V 


V' 


u" 

2\ 


— 


v' 

~2 


y 


+ 


v' y> 
1! 1! 


u" 




V n 




1 


v n-i 



?i\ n\ {71 — 1)! 1 ! 



+ 



v n ~ 2 y" 
(;/ — 2)! 2T 



. . . +v 



y 



To find y n , the wth derivative of u/v, in terms of the derivatives of 

from the 11 -J- 1 equa- 



te 
u and v. Eliminate y, — 

tions. We get 






' (« - 1): 

U V o 

u' v' 

— — V o 

1! 1! 

u" v" v'_ 

iT 2T 1 ! 



{?i -\- 1) rows 



SUPPLEMENTARY NOTES. 

Also, in particular, if u = i, we have 



455 



l -I)' 


V 


v o 


o . 










V 


IT v 


o . 


• • 




3- 


V" 7'' 
~2\ l\ 


v . 


• • 






n rows 





NOTE 4. 

Supplementary to § 56. 

To Find an Expression for the «th Derivative of a Function 

of a Function. 

Lets =y"(r), where j' = 0(-v). To find the «th derivative of 
z with respect to x. 

We have, by actual differentiation, 

A =fi?*, 

A" =f y "y* s -if,wi +/;/;'■ 

The law of formation of these first three derivatives of /"with 
respect to x shows that the wth derivative must be of the form 

A = AA + AA' + ■ ■ ■ + 4JS, (1) 

where the coefficients, A r , contain only derivatives of >' with respect to x 
and are therefore independent of the form of the function/". Conse- 
quently, if we determine A r for any particular function f we have 
determined the coefficients whatever be the function f Let then 



r\ 



Then in (1) we have 



- D%y - by = 



(y - i)~ 



A + 



y 



C— 0' 

Hence, when b = j>, we have 



1! 



A r _ x + A r . (2) 



^r = -A D&y - b) 



A 



which means that (y — 3) r is to be differentiated n times with respect 
to x and in the result y substituted for b. 



45 6 APPENDIX. 

This gives the nth derivative of y with respect to x in terms of the 
derivatives of/" with respect to y and those of y with respect to x, and 
is the generalization of the formula 

dx JKJ) dy dx 

We can give another form to (3), as follows. L,ety = b when 
x — a. Then 

y — b = 0(jv) - 0(a) = (* - a)z>, (4) 

where v stands for the difference-quotient 

<p (x) — 0(a ) 
x — a 
Apply Leibnitz's Formula to (4), and we have 

D%y - ?>) r = D n x {pc - ayv, 

But, Dt{x - a) r rr r(r -°i) . . . (r -/ + i)(* - a)-* 

= o when p > r, 

= o when p < r and .v = a. 

= r! when p = r and # = a. 

Therefore (3) becomes 

d\ n „, . Y^ 

dx, 



)A,)=jc,,/,,„(a"(*a^);.. « 



Notes 3 and 4 give some idea of the complicated forms which the 
higher derivatives of functions assume. 



NOTE 5. 

§ 64. Footnote. 

If a function /"(.r) and its derivatives are continuous for all values 
of a- in (nr, ft) except for a particular value a of x at which J\a) = 00 , 
then all the derivatives o{/(x) are infinite at a. 

Let ATj < x 2 < a. Then 

/(■g-/(-v,) = (.v 2 -.v I )/'(5), 

where B, lies between ^ 2 and x v Let a — x x be a small but finite 
number, and let x 2 ( = )a. Theny^.v.,) is infinite, andy^v,) is finite. 

.-. (*-*,}/"(£) = 00. 

Since a — x y is finite, f\£>) = °o ; and since/"'(c?) is finite if a — £ 
is finite, we must have a — £( = )o and 

/» = CC . 



SUPPLEMENTARY NOTES. 457 

In like manner we show that /"(a) — co , and so on. 
Corollary. If /[a) — co , then / '(a) = co , and also 

./i.v, 
becomes co when x = a. 

For, considering absolute values, if f \a) — co , then also 
l°gf( a ) — °° • By tne theorem established above, if \og/[x) is co 
when x = a, then 

also becomes co when x = a. 



NOTE G. 
Supplementary to Chapter VI 

On the Expansion of Functions by Taylor's Series. 

1. This subject cannot be satisfactorily treated except by the 
Theory of Functions of a Complex Variable. The present note is an 
effort to present in an elementary manner by the methods of the 
Differential Calculus a fundamental theorem regarding the elementary 
functions. 

An elementary function may be defined to be one which does 
not become o or co an infinite number of times in any finite interval, 
however small. Such functions are also called rational. 

A function _/*(.%■) is said to be unlimitedly differentiable at x when 
all the derivatives f r {x) of finite order are finite and determinate at 
x, We consider only those functions which are such that neither 
the function nor any of its derivatives become o or oo an unlimited 
number of times in the neighborhood of any value x considered. 

2. In the same way (hat a function of the real variable x may be 
o for an imaginary number p -f- iq t such a function may be oo for a 
complex number p -\- iq, where i = V—i. For example, the func- 
tion 

* (X) ~ </'(.v) 

becomes oo at p -\- iq if p -f- iq is a root of i/-(x) and not of <fi(x). A 
value of x at which F{x) is o or co is called a root or pole, respectively, 
of the function. It being understood that there are not an indefinite 
number of roots or poles in the same neighborhood.* 

* A point in whose neighborhood there are an infinite number of poles is called 
an essential singularity . An isolated pole is called a non-essential singularity. 



45 8 APPENDIX. 

The poles of a function, whether imaginary or real, enter into the 
results which we shall obtain. Wherever we use the word function 
in this note we mean a uniform function which has only roots and 
poles, but no essential singularity, and which is unlimitedly differen- 
tiable everywhere except at a pole. 

3. Theorem I. — \if{pc) is a one- valued, determinate, and unlim- 
itedly differentiable function at x, then the series 

o 

is absolutely convergent for all values of y less in absolute value than 

R = ^(x-pf + q*, 

where p -f- iq is the nearest pole of _/"(.*■), or any of its derivatives f r (x) t 
to the number x; and the series S is co for any value of y greater 
than R. 

4. Represent x, y by the coordinates of a point in a plane xOy. 
Then (see § 15, Ex. 9, 10): 

(1). At all points x,y at which 

y /-+«(*) 



/: 



n + 1 /«(*■) 

n —te 

S is absolutely convergent, and also 

£ »'■ 



< 1. 



£*-j 



(2). At all points x, y at which 



£-. 



n + 1 f«(x) 
S = co , and also 



l>llf 



/4/w 



5. It follows, therefore, that if 



/ 



Remembering that the modulus or absolute value of any number x -}- iy is 



l / 2 4- Vi that c 

\p + iq -x\ 



then of two numbers / x -(- ^ and/ 2 -f- ^2 that one is nearest jt for which we have 
the difference 

least. 



SUPPLEMENTARY X* >! 1 5. 459 



has a finite limit different from o, it is necessary that 



£-■ 



\ = V 



" + 1 /"(•>) 

6. Since at all points of absolute convergence of S 
and at all points of infinite divergence of S 



I 






the boundary between absolute convergence and infinite divergence 
of 6" is marked by the values of x, y which satisfy 

£t\™ = *• 

7. The locus 

£/"(*) = x, 

for an arbitrary and great value of n, will be a close approximation to 
the boundary line we seek. Differentiating, this locus has the differ- 
ential equation 

dy _ y f n +\x) 







dx n 


f\ 


X) 










--(■ 


] 

+; 


1 


y 


f n ^{x\ 


w 


tiich for n 


arbitrarily great gives 


, in 


the limit 


> 






dy 

dx ' 


1*. 








in 


virtue of 


r y f H+ 


\x) 




It 





Jjn+i f\x) ' ' 
on the boundary. 

8. Therefore the absolute value o(y is equal to the absolute value 
of a linear function of x, of the form 

y 1 = J k — x 1 2 , 
for all values of x andj^on the boundary. 

This is the equation of the family of boundary lines having the 
parameter k. These lines are fixed by the fact that whenever /"(.v) or 
f r (x), for any finite r, is 00 we have^ — o. 

9. If, therefore, f{x) — co when x — p, the corresponding bound- 
ary lines for a real pole p are the two straight lines 

f = (P ~ *)\ 
or y — x — p and y = — x -\-p. 



460 



APPENDIX. 



If /(-v) = co when x = p -j- *<7, then the corresponding boundary 
lines for a complex pole/ -f- ? !7 are tne two branches of the rectangu- 
lar hyperbola 

f = \P + *g - x\ 2 , 



or 



(-v - />)* 



+>=*• 



having for asymptotes 

jr = x — p and j' = — x -f- />. 

10. Therefore for any function having real and complex poles 
the boundary lines consist of pairs of straight lines crossing Ox at 45 
at the real poles and of right hyperbolae having as asymptotes similar 
straight lines crossing Ox at the real part of the complex pole. 

The vertices of the hyperbola corresponding to the pole/ -[- iq 
are/, ± q. 

11. The region of absolute convergence of 6' is that portion of 
the plane (shaded) such that from any point in it a perpendicular 
can be drawn to Ox without crossing a boundary line. The nearest 
boundary lines to Ox make up the boundary of the region of converg- 
ence of S. It consists of straight lines and hyperbolic arcs. 




Fig. 159. 



The boundary line of the region is symmetrical with respect to 
Ox. The ordinate at any point of this boundary line of converg- 
ence is the radius of convergence for the corresponding abscissa, and 
is equal to the distance of its foot from the nearest pole point. 

For any point on the boundary 



/ 



n _^ 1 f n {x) ' ' 



is less than 1 for any point inside, and greater than 
outside, the region. 



for any point 



SUPPLEMENTARY NOTES. 461 

12. If a function has two real poles », /)', and no pole between 
at, ft, the region of absolute convergence consists of a square between 

a and (3. If between a and (j their is an imaginary pole p -J- /r/ 
such that/ lies between ^ and /j, the imaginary pole has no inllu- 
ence on the region of convergence if 

[>--K« + /S)] 2 + ? 2 > \{a-P)\ 

If, however, 

the hyperbola^' 2 = (p — -*) 2 -{- q 2 cuts off a portion of the square of 
convergence. 

13. Theorem II. If f(x) is a one-valued, determinate, unlim- 
itedly differentiate function (having only a finite number of roots 
or poles in any finite interval), then 



Zi 



o 

for all values of x and y for which the series is absolutely convergent. 
That is, for all values oiy less in absolute value than the radius 



where/ -j- iq is the nearest pole of f(x) to x. Equation (1) is not 
true for any value oiy such that \y\ >R. 

Proof: The construction of the region of absolute convergence 
shows that from any point P in this region can be drawn two straight 
lines making angles of 45 with Ox to meet Ox without crossing or 
touching the boundary of absolute convergence. 

At any point x, y in the region of absolute convergence the 



series 



00 

-3 



/-+«(*) 



is absolutely 


convergent. 


But 




_dS _dS 

~ dx ~~ dy ' 


Hence 




= £(* + *> = 



(dx + dy), 

= o, if x -\-y is constant. 
Therefore all along the line x -\- y — c, in the region of absolute 
convergence, £ must be cons/an/. This line passing through any 



462 APPENDIX. 

point P in this region meets Ox without touching the boundary. At 
the point where x -\-y = c meets Ox we have_y = o, x = c, and 
S =/(c) =/(.v +.,■). 
Consequently all along any such line passing through the region 
of absolute convergence, and therefore at any point whatever in this 
region, we have 

o 

14. What is the same thing, 

00 

o 

for all values of xa.ndjy which make the series absolutely convergent.* 
If we make the investigation in the form (1), the regions consist 
of parallelograms on the line v = x as diagonal, and having for sides 
the straight lines 

X =p, X = 2V — p, 

corresponding to a real pole/, and hyperbolae 

or x 1 — 2xy -f 2pv = p l -f- f, 

corresponding to a complex pole/ -J- *<!■ 

15. Observations. — In the preceding investigation the object has 
been to point out as briefly as possible the salient points in the 
establishment of the theorems proposed. Details have not been 
entered upon. For example, we might discuss fully the behavior of 
the approximate boundary line 

£/v) = > 

at the zeros oiffx). There the curve has vertical asymptotes, but 
closes up on the asymptote as n increases. Also, f n {x) cannot have 
the same zero point for an indefinite number of consecutive integers 
n unless the function is a polynomial. 

Again, if at any assigned point x the derivatives are alternately 
o, the radius of convergence is fixed by 



/ 



'+i)j^7 } \ = \ Rl ' 



since for absolute convergence we must have ' 



/ 



'•{" + /"(•«) 



< 1. 



* It being understood that y is at a finite distance from any value of the variable 
it which the function is 00 . 



SUPPLEMENTARY NOTES. 463 

This simply means that there are two poles that arc equidistant 

from the value x. If the poles of a function are all real, if is im- 
possible for more than alternate derivatives to be zero continually. 

If there are more than two poles equidistant from x, then at least 
one must be complex. 

If there be three equidistant poles from x, then one must be real 
and two imaginary, /> ± iq, and conjugate. Then the derivatives at 
.v are o alternately in pairs and the radius of convergence there is 

R*\ = \£n(n +!)(»+ 2)^L- y 

and so on. 

Points x, equally distant from several poles, are the singular 
points on the boundary. Elsewhere, for three poles, we can always 
write 

f"( v) f n (x\ f n+J (x\ /~ M+2 ( y) 

-C+xX-+»)/^ - nf±±. ( . + l) ^.(. +8 )£_U , 

the limit of which is A 3 , and converges to the value R % at the singular 
point as a- converges to the x of such a singular point. The generaliza- 
tion of this is obvious. 



EXAMPLES. 

1. The region of absolute convergence and of equivalence of the Taylor's series 
of the functions tan x, cot x, sec x, esc x, consists of the squares whose diagonals 
are the intervals between the roots of sin x, cos x, respectively. 

2. In particular tan x is equivalent to its Maclaurin's series for all values of x 
in ) -\n. + \7t{._ 

Also for sec x in the same interval. 

cot x, esc x are equal to their Taylor's series in the interval )o, it{ , the base of 
the expansion being \%. 

x 

3. Expand by Maclaurin's series. 

v e* — I J 

Put y equal to the function. Then 

ye x — y — x. 
Apply Leibnitz's formula, and put x = o in the result. We have for deter- 
mining the derivatives of y at o, 

Making n = I, 2, 3, . . . , we find these derivatives in succession, and there- 
fore 

x x . B, „ B 9 , , B % 



= i-l + ^L X * -^L x *+^x* 

-5 ' -7 I A I 'fit 



e* — I 2 2! 4 

wherein B x = *, B 2 = fa B z = fa B x = fa B, = fa . are called 

Bernoulli's numbers. They are of importance in connection with the expansion 
of a number of functions. 

Since e ±in = — 1, the poles ± ire are the nearest values of x to o at which the 
function becomes 00 . The series is therefore convergent and equal to the function 
for x hi ) — 7t, -\- 7t. 



4^4 APPENDIX. 

4. Show that forx in ) — In, + £#(, 

?+! =3-- ttC* " I) + ~4V ( 24 - x > - iy ( 26 - ■>+ • 

either directly or from 

x x 7.X 



C x + I <?* — I *2* _ J 

5. Show directly from 4 that, for the same values of x, 

e x — 1 _ . , , V 2* — 1 i9„x 5 2 6 - I 

= B,x(2 l — I) — z - 5 ... 

6. Obtain the Maclaurin expansion 

sm(m sin-'x) - — r -f — j •* H 5 — ^ - -x 5 -\- 

and find for what values of x the equation is true. 
Put y = sin {m sin— ijr). 

. •• (1 — x 2 )y" — xy' 4- ni 1 }' = o. , 
Apply Leibnitz's theorem and deduce 

(1 - x 2 ) y(»+i) — (2« 4- i)jrj(«+i) + (w 2 — n 2 )y(») — o. 

v(«+i) y(«) 

. , (I _ ^) _ 2x{n + i} Z^ _ ( „2 _ ,„ 2) >_ = a 



/y(n + l) 
(* + *) jk^t |r= i^' 

(„2 _ w -2) _> _ / (« -f I) -f . (« -J- 2) ^ r I = I R 2 . 

we have 

(1 - x 2 ) - 2x R — R 2 = o, 

or R I = 1 x ± I. 

Therefore if jr is the base of a Taylor's series for y, the function is equal to the 
series in )x — 1, x -J- i(. If jr = o. the Maclaurin's series is equal to the function 
in )- 1. + i(. 

When x = o, the differential equation gives 

y(n+2) — („2 _ m 2 )y(g\ 
which gives the coefficients in the series. 

7. Treat in the same way cos {jh sin— *x). 

8. For what values of x is the Maclaurin's series corresponding to the function 
y in 

(1 - x 2 )y" - xy' — a 2 y = o 

equal to the function? 

W< >rk as in 6. The function is e a sin -1 *. 

9. In general, any function y satisfying a differential equation 

(I _j_ ax 2 )y("+^ -\-px(n + b)yV l + *) + q{n — c)(n — d)yi») = o, 

where <?, 6, r. d, p, q are any constant.-, is equal to its Taylor's series (base x) in 
the interval )x — R. x -(- R(, where R is the radius of absolute convergence, and 
R is the absolute value of the least root of the quadratic 

(1 4- ax 7 ) 4- px R 4- qR* = o. 

A large class of functions can be treated in this way. 



SUPPLEMENTARY NOTES. 465 

10. If u is a function of x having only a finite number of roots in a finite interval, 

find the region of equivalence of the function l/« with its Taylor's scries. 

Let y — l/u. Then yu= 1. Differentiate n times by Leibnitz's formula. Then 

y n " -f ^1;"-'"' + Cn,2\ ,H - 2 "" + . . . + «" = o. 
Divide by y n , and make n — 00 . Then, p being the radius of convergence, 
we have, if u — <p(x), 

<P{x) + ft ~ <p'(x) + £ &'(x) + . . . = a 

But this series is nothing more than <p(x -\- p). 
Therefore x -j- p must be a root of 

0(* + P) = o. 
Consequently 

* -f- p - k, 
or p = k — x, 

where k is the nearest root of <p(x) to .r, the base of the expansion. 

NOTE 7. 

Supplementary to Note 6. 

1. While, in this book, we are not interested in functions of a 
complex variable z = x -f- iy, it is instructive and interesting to con- 
sider the treatment of a function /[z) after the method of Note 6 for a 
function of the complex variable z = x -\- iy. 

We assume XhdXflz) is one-valued, unlimitedly differentiable with 
respect to z at all values of z in the finite portion of the plane except 
at poles o{/{z), which are, we assume, the only singularities the func- 
tion has. 

2. Let z = x -f- iy, C = x' -j- iy'. The series 

00 

o 

is absolutely convergent (when the series of absolute values of its terms 
is convergent) for all values of z and C which satisfy 

The series 6" is c* when these limits are greater than 1 . 
The boundary conditions are 



! = |i. 



Therefore for n arbitrarily great the boundary is arbitrarily near 
a and /? being arbitrary constant real numbers. 



466 



APPENDIX. 



From the first of these equations, we have 
dz_ s /* +I (C) 

« n /«(£) ' 

= — £ ?/3 , when n = oo. 

z = k — e l>, 

k being an arbitrary constant. But if/ is a pole of/js), then 2 = o 
when £ =p. 

Hence o = k — £ 7/3 />, or /£ = e^p. 

Therefore, corresponding to any assigned £, the boundary corre- 
sponding to the pole/ is fixed by 

which is a circle about the origin in the z-plane with radius 

£=\P-Z\, 

since /? is arbitrary. 

3. lip is the nearest pole oif(z) to C, then for all values of z for 
which 

|»|<^ = |/-C| 

the series is absolutely convergent, and is infinite for any value of z 

\i\z\>R. 

4. Put £ = z + C. Then z = B, - Q. 

The series 

o 

is absolutely convergent at all points 
£, inside the circle Cdescribed about 
x C as a center with radius 

R=\Z-P\, 

FlG - l6 °- p being the nearest pole of f{z) to C. 

For any assigned value of B, in this circle the series S is con- 
stant with respect to C, since 




r ! 



« I 



and this is o when ?i = 00 . 

Now we can always move C np to <? along the straight line join- 
ing them, the series £ remaining constant in value. But when 
£ = £, we have 



00 

E 



0- 



?! ! 



/"(0 =/**)• 



SUPPLEMENTARY NOTES. 467 

Therefore* this equality is true for all values of £, Z which make 
the series absolutely convergent, i.e., at any point inside the cir- 
cular boundary corresponding to any assigned Q and the nearest pole 
p oif(z), described about £ as center with radius of absolute con- 
vergence, 

* = |/-C|. 

NOTE 8. 
Supplementary to Note 6. 

Pringsheim's Example of a Function for which the Maclaurin's 
Series is absolutely Convergent and yet the Function and 
Series are different. 
Let 



Z, r ! l + * 



A*)= ) - ] r \ > (r) 



A and a being positive constants, a > 1. This function is one- 
valued, finite, continuous, and unlimitedly differentiate for all finite 
values of the real variable x. It has, however, infinitely many com- 
plex poles 

±—#r-> r=i, 2, 3, ... 

an infinite number of which are in the neighborhood of x = o, which 
is therefore an essentially singular point. 

For the nth. derivative oif(x) we find (i = -f- V ' — 1) 

o 

At x = o, 

f(o) = e-\ 

f am + *(o) = o, 

/ 2 "'(o) = (— i)" ! (2m)\ e~^ 2m . 

Therefore the Maclaurin's series is 

00 
X - ^ 7 x 2r 

■ s =2 ( - ir ^' (2) 

o 

This series is absolutely convergent for all finite real values of x. 

*This problem was first solved by Cauchy, by means of singular integrals. 
See any text on the theory of functions of a complex variable. 






4 68 



APPENDIX. 



Now let A < It |.*| < 

• •• A*)> 



I -f- X 2 I -J- a 2 *'" 
and 6" < *-*. 

In particular, let x = a - *. 



> 



i -f~ -*' 2 i + # 2 -* 2 







Fig. i6i. 



/(*"*) > 



> 




i -(- i/fl i -\- a <z-f-i 
.-. /(a - *) > e~ l > .S when x = a~± y 

when — ■ — > e -i or a > . 

a -\- i = ■> = e — i 

The function /*(*) and the series £ 
are different. 

In the figure the solid line is the 

curve y =f(x), the dotted line the curve 

-a; y — S, constructed with exaggerated 

ordinates, for the values X = log 2, 

a = 2.* 



NOTE 9. 

Supplementary to § 118. 

Riemann's Existence Theorem. 

Any function f{x) that is one-valued and continuous throughout 
an interval (a, b) is integrable for that interval. 

Let the numbers x xt x 2 , . . . , x n _ 1 be interpolated in the inter- 
val {a, b) taken in order from a = x Q to b =. x n . 

We have to prove that the sum of the elements 



S H = 2/{Zr)(Xr ~ *r-i) 



(I) 



converges to a unique determinate limit, when each subinterval con- 
verges to zero, whatever be the manner in which the numbers x r are 
interpolated in (a, b). 

I. The sum S n must remain finite for all values of n. For f(x) 
is finite, and if M and m are the greatest and least values oif(x) in 

m{b — a) < £„ < Af(£ — a). 

Also, since /*(*) is continuous, there exists a value B, in (a, <$) at 
which 

S. = (* - «)/(*), (2) 

/"(<?) being a value of/"(.r) between w and M. 



* For further information on this subject, see papers by Pringsheim, Math. 
Ann. Bd. XLII. p. 109. Math. Papers Columbian Exposition, p. 288. 



SUPPLEMENTARY NOTES. 469 

II. Interpolate in the rth subinterval of (1), in any manner, 
n' r — 1 values .v,' , . . . , x n ,_ l of x. Then, as in I, 

S K - y(OiK - -<~ J = (*r - *r-i)AZr)> (3) 

where B,' r is some number in the subinterval [x Ti x r _ 1 ). 

Form similar sums of elements for each of the n subintervals of 
(iV Let/ = n[ -f- . . . -f- »*. Add the « sums of elements such 
as (3). 

Hence 

s f = s„ i +... + s„, n , 

I 

= i (x r - *,-,)/(«). (4) 

I 

This is a new element sum containing/ > « elements, which is 
to be regarded as a continuation of (1) by the interpolation of new 
numbers in each subinterval of (1). 

Subtracting (4) from (1), we have 

s. - s, = i[/&o -/<«)](*, - -v,-,)- 

Let 6 be the greatest absolute value of the difference between the 
greatest and least values of f{x) in the subinterval (x r — x r _ 1 ), 
r — 1, . . . , n. Then, since f{z r ) and f{£' r ) are values of f(x) in 
{x r , x r _ I ) ) 

S n -.S,\<\62(x r -x„), 

I <!*(*-*), (5) 

for all values of the integer p, however great. But when each sub- 
interval converges to o, then fi( = )o, since f(x) is continuous, and 
at the same time n = 00 . 

Therefore, by the definition of a limit, S u converges to a limit 
when n = 00 . 

III. To show that the limit of S n is wholly independent of the 
manner in which the interval (a, b) is subdivided : 

Let there be an entirely different and arbitrary interpolation 
.r/, . . . , x' m _ x . Consider the element-sum 

m 

S' m = 2/[Zr){.K - <-:)• (6) 

1 

Interpolate in {a, b) the numbers 

x \ j • ■ • J X n— 1 > x i > • • • > X m—\ i 

occurring in (1) and (6), thus dividing (a, b) into m -f- n intervals. 



47° APPENDIX. 

Interpolate in each of these m -\- n intervals new numbers, thus 
dividing (#, b) into m -f- n -\- p subintervals. Form the element- 
sum S m + n + p corresponding to these subintervals. 

Then, by II, S n and S m + n + * converge to the same limit. In 
like manner S m and S m + M + P converge to the same limit. Therefore 
S n and S m converge to a common limit. The uniqueness of the 
limit of (i) under any subdivision whatever of (a, b) is demonstrated. 

This theorem gives the means of denning analytically the area and 
length of a curve, and the volume and surface area of a solid. 

NOTE 10. 

Supplementary to § 135. 

Formulae for the Reduction of Binomial Differentials of the form 

x a (a 4. bx n ) y dx. 
Put y — a + bx n . Then 

Dxyv = ax a -y y -f- nybxf^- M - i yr- 1 i 

— aax a -i\n-T- -j- (a a -j- ny)bx aJ r^- x y*- x > (1) 

= (a -f- ny)x a —[yy — anyx a —]yy— 1 . (2) 

In (1), put a = m— n+i, y=p + i, then 

Bx m - n+i y +l =a(m — n+ i)x m - n y -f- (np + m + i)bx m yt (A) 
In (2), put a = m-\-i, y = p, then 

Dx m+ y = (np + m+i )x m y p — anpx m y*-\ (B) 

In (1), put or = wz -I- 1, y=^-fi, then 

D x ™+yfi+i — a{m + i)* w y -f {np -f- w + « + i)^ w+ V- (C) 
In (2), put « = w -f 1, y=p+i, then 

jr^w+yn = ( H p + m + n + i)x" ! yt +l — an(p + i)x m yK (D) 
Integrating the formulae (A), . . . , (D), we have the formulae of 
reduction, where y = a -{- bx n : 

^=t£^ (a) 

f xmyPdx = * W ^ +I _ (»/ + » + »+ i)* />y& (C) 

J *y ax (m+i)a (m + i)a J K > 

^'W* = h , , — r — / x m yt +1 dx. (D) 



SUPPLEMENTARY NOTES. 



47 



NOTE 11. 
Supplementary to § 165. 

H> =/(- r ) be represented by a curve, and y, By, B\y are uniform 
and continuous, then we can always take two points P and P x on the 
curve so near together that the curve lies wholly between the chord 
and the tangents at P and P y 

Let x, y be the coordinates ofP, and 
X, J^those of P', any point on the curve 
between P and P v 

The tangent at P has for its equation 
r t =/(.v) + (A' - x)/\x). 

At any point x, y of ordinary posi- 
tion, not an inflexion, the difference 
between the ordinate to the curve and 
the tangent is 

AX)-r l = {X - x) ' i /'\S), (1) 




where £, is some number between x and A^. We can always take X 
so near to x that /"{£) keeps its sign the same as that oif"\x) for all 
values of B, in (x, X). Therefore the difference (1) keeps its sign 
unchanged in (x, X) or the curve is on one side of the tangent, for 
this interval. 

The equation to the chord PP 1 is 

r. =/0) + (A - x)/'{s x ), 

where/X^i) is the slope of the chord PP V The difference between 
the ordinates of the curve and chord is 

/(A) - ¥ c = (A - x)[f\8) -/'(£,)]. (2) 

Let x l be so near x that/"^^), f'(£i^ have the same sign as/\x). 
Then this difference (2) keeps its sign unchanged for all values of A 
in (x, x x ). It can now be easily shown that (2) and (1) have opposite 
signs, and there can always be assigned a numbers* so near x that the 
curve PP X lies wholly in the triangle formed by the tangents at P, P x 
and the chord PP V 

NOTE 12. 

Supplementary to § 226, IV. 

Proof of the Properties of Newton^ Analytical Polygon. 

I. Let there be any polynomial in x and y, such as 



/ = w + 



?< 



+ A,„x*'"f 



(0 



wherein the exponents a, ft of each term satisfy the linear relation 

a a + bfi — c, (2) 

c being taken a positive number. 



47 2 APPENDIX. 

Let f be arranged according to ascending powers of y, so that 
fi x < ft % < . . . Then 

= x**[a i + A m (yx~y + . . J, (3) 

r^V'U*"'-- V • • • G'*~"- fa-ftM- (4) 

where k x , . . . , kp _p x are the roots of the equation in / = yx a , 

A i + A /*-K' + . . . +Aj--* ==o. 

Therefore the locus of f = o consists of x = o, y = o, and the 
parabolic curves 

r = K x >. (r = i, . . . . , § m - /»,). 

b_ 

2. In (3), let_y = &* a , /£ being constant. Then 

b b 

f=A 1 x ai Z? i x 8l « + A 2 x a *A fi *x fi >«-{- . . . , 
= Ajf l x a * +fii 7+AJl! it x at+fi *T+ . . . , 
= x T {a i #*+A/*+ . . .) 

= ^T.r a , A' being constant. 

3. Let /' be a function A'xr-'yfi' , or the sum of a finite number of 
such functions, such that the exponents a', ff of each term satisfy the 
linear equation 

aa' + bfi' = <;'. 
_*_ 
Then, as in 2, \ety = Ar a , and we have in the same way 

c' 

f = K / x a i 

K' being a constant. 

4. Let a, b and c, c' be positive numbers. 
Then 

/' - K 'x^ 
f ~ K > 

where je andj> satisfy y 1 = kx 5 . 
(1). If c' > c, then 

//' 
— — = o, when .r( = )o, j/( = )o. 



SUPPLEMENTARY NOTES. 



473 



£>. 



(2). If c' < c, then 

/' 1, 

~7T = o, when x = 00 , _^ = cc . 

5. We are now prepared to prove § 226, IV, (1), (2). 
Let *\x,y) = 2C r x>jft = o. 

(1). Let/ represent that part of /"which corresponds to a side of 
the polygon as prescribed in § 226, IV, (1), and F' represent the 
remainder of F. Then 

F=/+F', 
F F' 

7=1 + 7* 

Through each point corresponding to terms in F' draw a line 
parallel to the side corresponding to/* 
Then by 3, (1), we have 

/F . FF' 

? =I ' smce iz =0 * 

when x{ — ) o, y( = )o. 

Therefore in the neighborhood of the origin F = o and /= o 
are the same. 

But the form of/" = o in the neighborhood of the origin is that of 
a parabola 

y* = kx*. 

Hence F = o goes through the origin in the same way as does 
f = o, whose form is that of a parabola of type y* = &e*. 

(2). Let /"represent that part of F corresponding to a side of the 
polygon as prescribed in § 226, IV, (2), and F' the remainder of F. 

F F' 

Then = T + 

Draw parallels to the side corresponding to/*, through all points 
corresponding to terms in F' '. 
Then by 3, (2), we have 

/F . FF' 

7 =I ' since i/ =o ' 

when x = 00 , y = 00 . 

Therefore F = o and/*= o pass off to 00 in the same way. Also, 
/" = o passes off to 00 , as does a parabola of type,/* = kx b . 

Note. — The same process can be extended to surfaces, using a polyhedron in 
space. The part of the equation corresponding to a plane face such that there are 
no points between that face and the origin gives the form of a sheet of the surface 
at the origin. Likewise the part corresponding to a plane face such that no point 
lies on the side opposite to the origin gives the form of a sheet at 00 . The plane 
faces in each case cutting the positive parts of the axes. 



INDEX. 



[The numbers refer to the pages, ,] 



Absolute number, 2 
Anticlastic surface, 360 
Appendix, 451 
Archimedes, 

spiral of, 117, 161 

area of spiral, 234 

length of spiral, 248 
Areas of Plane Curves, 

rectangular coordinates, 226, 396 

polar coordinates, 233, 397 
Asymptotes, 

rectilinear, 121 

to polar curves, 125 
Auxiliary equation, 443 
Axes, of a conic, 323, 325 

of a central plane section of a coni- 
coid, 328 

Base of Expansion, 88 
Bernoulli. 

definite integral by series, 222 

differential equation, 424 
Binomial differentials, 193, 470 
Binomial formula, 67 
Binormal, 378, 379 
Bonnet, 131 
Boundary of region of convergence, 460 

Cantor, definition of number, 5 
Cardioid, 118, 163 

area, 234; length, 248 

surface of revolute, 261 

volume of revolute, 401 

orthogonal trajectory of, 438 



Catenary. 

normal-length, 1 16 

radius of curvature, 134 

center of curvature. 146 

curve traced, 152 

area, 228 ; length, 245 

volume of revolute, 258 

surface of revolute, 261 

differential equation, 446 
Cauchy, 

theorem of mean value, 79, 87, 222 

theorem on undetermined forms, 93 

on expansion of functions, 467 
Caustic by reflexion, 390 
Circle, 

area, 227, 234 

length of perimeter, 246 
Circle of curvature, 

for plane curves, 100, 134 

for space curves, 379 
Cissoid, 

tangent and normal, 115 

subtangent, 116 

curve traced, 151 

area, 229 
Clairaut's equation, 429 
Coordinates of center of curvature, 133 
Computation of, 

e, 84; logarithms, 86; it, 89 
Concavity and Convexity, 127 
Concavo-convex, 128 
Conchoid of Nicomedes, 160 
Concomitant, 312 
Convexo-concave, 128 

475 



476 



INDEX. 



Cone, 

volume of, 257, 266 

equation of, 349 
Conic, center of, 331 
Conicoid of curvature, 360 
Conjugate points. 334 
Connectivity, law of, 19 
Conoid, volume of, 266 
Consecutive numbers, 7 
Constant, 4 
Contact, 

of a curve and straight line, 127 

of two curves, 130 
Continuity, 

theorem of, 23 

of functions, 278 
Continuum, 3 
Convergency quotient, 14 
Cubical parabola, 135, 150 
Curvature, 130 

radius of, 133 

circle of, 134 

of surfaces, 365 

measure of, 370 

spherical, 381 
Curve tracing, 147, 340 
Curves in space, 375 
Cusp, 151, 155, 336, 337 
Cusp-conjugate point, 335 
Cusp-locus, 435 
Cycloid, 

tangent to, 114 

curve traced, 163 

area, 232 

length, 252 

surface of revolute, 261, 263 

volume of revolute, 262, 263 
Cylinder, equation of, 348 

Decreasing function, 74 
Definite integration, 215 
Degenerate forms of differential equa- 
tion of second order, 440 
Degree of differential equation, 409 
Descartes, 26 
Develop ible surfaces, 374 



Devil, 161 
Difference, 

of the variable, 35 

of the function, 35 

quotient, 36 
Differential, 55, 63 

quotient, 55, 64 

coefficient, 55 

relation to differences, 57 

total, 294 
Differentiation of, 

logarithm, 41 

power, 42 

sum, 43 

product, 44, 69 

quotient, 45, 454 

inverse function, 46 

trigonometrical functions, 44, 45, 46 

circular functions, 48 

exponentials, 49 

function of a function, 49, 70, 453, 

455 

implicit function, 66, 296 

function of independent variables, 
282, 306 

under the integral sign, 391 
Differential equations, 

first order, 409 

second order, 439 
Discriminant equations, 434 
Double points, 334 
Double integration, 396 
Dumb-bell, 160 

Edge of envelope, 389 
Elements of curve at point, 147 
Elliott's theorem, 236 
Ellipse, 

tangent and normal, 113 

subnormal, 1 15 

radius of curvature, 125 

evolute, 144, 154 

area, 228, 233 

arc length, 246 

length of evolute, 246, 251 

normal to, 331 

orthogonal trajectory, 433 



[NDEX. 



477 



Ellipsoid, volume of, 268, 399 
Elliptic functions, 208 
Elliptic paraboloid, 268, 280 
Envelopes, 

of curves, 138, 343, 435 

of surfaces, 385 
Epicycloid, 164, 248 
Equiangular spiral, 118, 162, 234 
Equilateral hyperbola, 

evolute of, 146 

area of sector, 228 
Euler's theorem on curvature, 366 
Eulerian integral, 217 
Evolute of a curve, 144 

length of, 250 
Exact differential, 416 
Exponential curve, 150 

Family of curves, 138 
Finite difference-quotient 

/(*) -A*) 

x — a 

«th derivative of, 138 
Folium of Descartes, 

tangent, 1 13; asymptote, 122 

traced, 159; area, 232, 234 
Fort, 18 

Fresnel's wave surface, 390 
Function, 

definition, 19, 273, 274 

explicit, implicit, 19, 277 

transcendental, 20 

rational, irrational, 20 

symbolism for, 21 

uniform, one- valued, 21 

continuity of, 22, 278 

difference of, 35 

derivative of, 36 

difference-quotient of, 36 

increasing, decreasing, 74 
Function of a Function, 

geometrical picture, 453 

«th derivative of, 455 

Gamma functions, 217 



Gauss, 

theorem on areas, 241 

theorem on curvature, 372 
Geodesic line, 384 
Groin, volume of, 267 

surface of, 408 

Harkness, 451 
Helix, 376, 384, 405 
Holditch's theorem, 238 
Homogeneous, 

coordinates, 339 

differential equation, 414, 431 
Horograph, 372 
Hyperbola, 

tangent, 113; asymptotes, 122 

radius of curvature, 137 

area, 228, 233 

orthogonal trajectory, 432 
Hyperbolatoid, volume of, 269 
Hyperbolic sine, cosine, 29 
Hyperbolic spiral, 

traced, 162; subtangent, 117 

area, 234 ; length, 248 
Hyperbolic paraboloid, volume, 399 
Hyperboloid of revolution, volume, 258 
Hypocycloid, 

tangent, 113; evolute, 146 

traced, 154, 164 ; area, 229 

length, 246 

volume of revolute, 258 

surface of revolute, 261 

Increasing function, 74 
Indicatrix of surface, 361 
Infinite, infinitesimal, 2, 7 
Inflexion, 128 

Inflexional tangent, 332, 355 
Integer, definition, I 
Integral, definition, 165 

indefinite, 173; definite, 215 

fundamental, 173 
Integration, definition, 167 

by transformation, 178 

by rationalization, 182 

by parts, 183 



478 



INDEX. 






Integration by partial fractions, 185 

under the /'sign, 393 
Integrating factor, 420 
Interval of a variable, 4 
Intrinsic equation of curve, 251 

of the catenary, 252 

of the involute of circle, 252 

of the cycloid, 253 
Illusory forms, 95 
Inverse curves, 161 
Involute of a curve, 144 

Jacobi's theorem on areas, 241 

Lacroix, 335 
Lagrange, 

theorem of mean value, 78 

differential equation of, 430 

interpolation formula, 241 
Leibnitz, 

«th derivative of product, 69 

symbol of integral, 170 

linear differential equation, 423 
Lemniscate, 99 

traced, 156, 159 ; area, 234 ; length, 
253; area revolute, 263 
Lengths of curves, 

plane curves, 243, 247 

curves in space, 404 
L'Hopital's theorem, 94 
Limit, 

of a variable, 7 

principles of, 7 

theorems on, 8 

of (I + 1/2) 2 , 16 

of integration, 167 
Li ma 9011, area of, 239 
Linear differentiation, 293, 301, 307 
Linear differential equation, 443 
Line of curvature, 384 
Logarithmic curve, 

traced, 150 ; length, 246 
Logarithmic spiral, 

area, 234 ; length, 248 
Lpxodrone, 384 



M 



u l.iunn 5 



series, 83, 467 



Maximum and Minimum, 103 
independent variables, 314 
implicit functions, 321 
conditional, 322 

McMahon, 416 

Mean Value, 

theorem of, 76, 218 
formula by integration, 220 
for two variables, 309 

Mean Curvature, 371 

Meunier's theorem, 366 

Modulus of a number, 3, 458 

Morley, 451 

Neil, 245 

Neighborhood, 7, 278 
Newton, 

binomial formula, 67 

radius of curvature, 135 

rule for areas, 240 

analytical polygon, 340, 471 
Nodal point and line, 362 
Node, 156, 337 
Node-locus, 435 

Non-exact differential equation, 418 
Normal, 

to a curve, 114, 115, 330 

to a surface, 358 
Normal plane, 376 

Oblate spheroid, 

volume, 258; surface, 262 
Omega, 2 

Order of differential equation, 409 
Ordinary point, 

on curve, 329; on surface, 352 
Orthogonal trajectories, 432 
Osculation, 132 
Osculating plane, 377 

Parabola, 

tangent, 113; subnormal, 116 
radius of curvature, 134 
evolute, 144; area, 228, 234 
arc length. 245. 248 
length of evolute, 251 
orthogonal trajectory, 432 



INDEX. 



479 



Paraboloid of revolution, 
volume, 258; surface. 260 

Parameter, 138 

Partial derivatives, 282 

Pedal curve, 239 

Plane, equation to, 347 

Planimeter, 239 

Pole of a function, 457 

Primitive of a function, 168 

Principal, 

sections of surface, 365 
radii of curvature, 365, 368 
normal, 378 

Pringsheim, 87, 467 

Probability curve, 

traced, 151; area, 395 

Prolate spheroid, volume, 257 

Pseudo-sphere, 

volume, 258; surface, 261 

Pursuit, curve of, 446 

Quotient of functions, 
nih derivative, 454 

Radius of convergence, 88 
Radius of curvature, 

plane curves, 100, 133, 250 

at point of inflexion, 135 

for surfaces, 365 

for space curves, 379, 389 
Real number, 3 
Reciprocal spiral, 1 17. 126 
Revolute, definition, 255 

volume, 256 ; surface, 259 
Riemann's existence theorem, 468 
Roche, 222 
Rolle's theorem, 75 
Root of a function, 457 

Saddle point, 336 
Scarabeus, 160 
Schlomilch, 222 
Semi-cubical parabola, 245 
Sequence, 14 
Singular points, 

on a curve, 158. 333 



Singular points on a surface, 352, 362 
Singular solutions of differential equa- 
tions, 433 

Singular tangent plane, 362 
Singularity, essential and non-essential, 

457 
Solution of differential equations, gen- 
eral, particular, complete, 410 
by separation of variables, 410 
when M and N are of first degree, 

415 

by differentiation, 426 

when solvable for y, 428 

when solvable for x, 430 

when solvable for/, 431 
Specific curvature, 371, 372 
Sphere, 

volume, 257, 400; surface, 260, 403 
Spherical curvature, 381 
Steradian, 371 
Stewart, 238 
Stirling, 83 

Straight line, equations, 348 
Subtangent, subnormal. 115, 116 
Successive differentiation, 62 
Surface, definition, 349 

general equation, 349, 361 

of solids, 255, 402 
Synclastic surface, 360 

Table of derivatives, 52 
Table of integrals, 176 
Tac-iocus, 435 
Tangent, 

to plane curves, 112, 116, 330 

length, 115 

to space curves, 375 
Tangent line to surface. 350 
Tangent plane, 35 1 
Taylor's series, 82, 86, 87. 221. 457. 

467 
Tortuosity, 380, 382 
Torse (developable surface), 374 
Torus, 

volume, 259; surface. 261 

tangent plane to, 364 









480 

Total, 

derivative, 291, 294 

differentiation, 290 

differential, 294 
Tractrix, 

tangent-length, 119; area, 235 

arc length, 246 

volume of revolute, 258 
Trajectory, 431 
Transcendental function, 83 
Triple, 

point, 337; integration, 398 
Trochoids, 164 

Umbilic, 361, 369 
Undetermined forms, 92 



INDEX. 



Undetermined multipliers, 

applied to maxima and minima, 323 
applied to envelopes, 343, 388 

Undulation, point of, 333 

Variable, definition, 4 

difference of, 35 
Volumes of solids, 255, 398, 400, 401 

Weierstrass, 5 

example of derivativeless function, 

45 1 
Witch of Agnesi, 

tangent, normal, 115 
traced, 152; area, 228 
volume of revolute, 258 

Zero, 2 



MATHEMATICS 



Evans's Algebra for Schools. 

By George W. Evans, Instructor in Mathematics in the English High 
School, Boston, Mass. 433 pp. i2mo. $1.12. 

Aside from a number of novelties, the book is distinguished 
by two notable features : 

(1) Practical problems form the point of departure at each 
new turn of the subject. From the first page the pupil is put 
to work on familiar material and on operations within his 
powers. Difficulties and novelties arise in a natural way and 
in concrete form and are met one at a time, and he is led to see 
the need for each operation and preserved from regarding 
algebraic processes as a species of legerdemain. 

(2) The book contains nearly 3,500 examples, none of which 
are repeated from other books. The exercises are graduated 
according to difficulty and are adapted in number to what ex- 
perience has shown to be average class needs. Problems are 
carefully classified with reference to the several types of 
equations arising from them, and the pupil is specially drilled 
upon typical forms (as, for example, " the clock problem," " the 
cistern problem," "day's work problem," etc.) and upon 
generalized forms. 



Paul H. Hanus, Professor in 
Harvard University : — The author 
has certainly been successful in 
presenting the essentials of ele- 
mentary algebra in a thoroughly 
sensible way as to sequence of 
topics and method of treat- 
ment. 

C. H. Pettee, Professor in the 
N. H. College of Agrictilture : — I 
have actually become tired look- 
ing over algebras, geometries, and 
trigonometries that have no ex- 
cuse for existence. Hence it is 
with real pleasure that I have 
examined Evans's Algebra for 
Schools. The author evidently 
knows what a student needs and 
how to teach it to him. 



E. S. Loomis, Cleveland {Ohio) 
West High School : — To pass 
gradually from arithmetic to all 
gebra. to bridge that intellectua- 
chasm in the minds of many, 
is no little thing to do. Evans 
has done it more nearly than 
any other author I have read. 
I like his scheme of models, 
but above all I like his coor- 
dinating algebra and the other 
sciences. I wish I could teach the 
book, it is so full of good things. 

Jas. E. Morrow, Principal Al- 
legheny {Pa.) High School:— I find 
more to commend in this algebra 
than in any book on the subject 

since the publication of 's in 

1869. 

32 



Mathematics 



Gillet's Elementary Algebra. 

By J. A. Gillet, Professor in the New York Normal College, xiv + 412 
pp. i2mo. Half leather, $1.10. With Part II, xvi 4-512 pp. 121110. 
$i-35- 

Distinguished from the other American text-books covering 
substantially the same ground, (1) in the early introduction of 
the equation and its constant employment in the solution of 
problems ; (2) in the attention given to negative quantities and 
to the formal laws of algebra, thus gaining in scientific rigor 
without loss in simplicity ; (3) in the fuller development of 
factoring, and in its use in the solution of equations. 



James L. Love, Professor in 
Harvard University : — It is un- 
usually good in its arrangement 
and choice of material, as well as 
in clearness of definition and ex- 
planation. 

J. B. Coit, Professor in Boston 
University : — I am pleased to see 
that the author has had the pur- 
pose to introduce the student to 
the reason for the methods of al- 
gebra, and to avoid teaching that 
which must be unlearned. 



F. F. Thwing, Manual Train- 
ing High School, Louisville, Ky. : — 
Two features strike me as being 
very excellent and desirable in a 
text-book, the prominence given 
to the concrete problems and the 
application of factoring to the so- 
lution of quadratic equations. 

J. G. Estill, Hotchkiss School, 
Lakeville, Conn.: — The order in 
which the subjects are taken up 
is the most rational of any algebra 
with which I am familiar. 



Gillet's Euclidean Geometry. 



By J. A. Gillet, Professor in the New York Normal College. 
i2mo. Half leather, $1.25. 



436 pp. 



This book is " Euclidean " in that it reverts to purely geo- 
metrical methods of proof, though it attempts no literal repro- 
ductions of Euclid's demonstrations or propositions. Metrical 
applications and illustrations of geometrical truths are inter- 
spersed with unusual freedom. " Originals " are made an 
integral part of the logical development of the subject. 



Percy F. Smith, Professor in 
Yale University : — The return of 
the "spirit of Euclid" should be 
much appreciated, and it will be 
interesting to watch the workings 
in the classroom of the two alter- 
native methods of Book V. Con- 
sistency and rigor are carefully 



maintained in both works, and I 
shall take great pleasure in using 
and recommending them. 

E. L. Caldwell. Morgan Park 
Academy * III.:— I find in them the 
best results of modern research 
combined with rigid exactness in 
definition and demonstration. 



34 



Mathematics 



Keigwin's Elements of Geometry. 

By Henry W. Keigwin, Instructor in Mathematics, Norwich (Ct ) Free 
Academy. iv-+-227pp. i2mo. $1.00. 

This little book is a class-book, and not a treatise. It cov- 
ers the ground required for admission to college, and includes 
in its syllabus the stock theorems of elementary geometry. It 
is, however, out of the common run of elementary geometries 
in the following particulars : 

i. The early propositions, and a few difficult and funda- 
mental propositions later, are proved at length to furnish 
models of demonstration. 

2. The details of proof are gradually omitted, and a large 
part of the work is developed from hints, diagrams, etc. 

3. The problems of construction are introduced early, and 
generally where they may soon be used in related propositions. 



Oren Root, Professor in Hamil- 
ton College, N. Y. : — I like the 
book, especially in that it gives 
" inventional geometry" while 
giving the fundamental propo- 
sitions. Geometry is taught very 
largely as if each proposition were 
an independent ultimate end. 
Pupils do not grasp the interlock- 
ing relations which run on and on 
and on unendingly." Mr. Keig- 
win's book, compelling pupils to 
use what they have learned of re- 
lations, must help to prevent this. 

C. L. Gruber,/^?. Normal School, 
Ktttztown ; — The method of the 
book is an excellent one, since it 
gradually leads the student to de- 
pend in a measure upon himself 
and consequently strengthens and 
develops his reasoning powers in 
a manner too often neglected by 
teachers of the present day. It 
gives neither too little nor too 
much. 

W. A. Hunt, High School, Den- 
ver, Colo. : — It does not do for the 
pupil what he should do for him- 
self. With strong teaching, the 
book is just what is needed in 
preparatory schools. 



Miss Emily F. Webster, State 
Normal School, Oshkosh, Wise: — 
At the first I looked upon the 
book as very small, but I now con- 
sider it very large, for it is per- 
fectly packed with suggestions 
and queries which might easily 
have extended the book to twice 
its present size had the author 
seen fit to elaborate, as so many 
authors do ; but in not doing so 
lies one of the finest features of 
the book, as much is thus left for 
the student to search out for him- 
self. The original exercises are 
fine and in some cases quite un- 
usual. The figures are clear and 
the lettering is economical, some- 
thing which is by no means com- 
mon, and much valuable time is 
wasted by repeating unnecessary 
letters in a demonstration. Dem- 
onstrations are made general, 
which is an advantage, for it is 
often difficult to induce pupils to 
do so when the author has failed 
to set them the example. 

George Buck, Dayton (O.)High 
School . — I am highly pleased with 
it and commend its general plan 
most heartily. 



Mathematics « 



Newcomb's School Algebra. 

By Simon NEWCOMB, Professor of Mathematics in the Johns Hopkins 
University, x + 294 pp. i2mo. 95 cents. lAVy, 95 cents. Answers, 10 
cents.) 

Newcomb's Algebra for Colleges. 

By Simon Newcomb, Professor in the Johns Hopkins University. Revised' 
xiv + 546pp. i2mo. $1.30. {Key, $1.30. Answers, 10 cents.) 

Newcomb's Elements of Geometry. 

By Simon Newcomb, Professor of Mathematics in the Johns Hopkins 
University. Revised, x -+- 399 pp. i2mo. $1.20. 

Newcomb's Elements of Plane and Spherical Trigo- 
nometry. (With Five-place Tables.) 

With Logarithmic and other Mathematical Tables and Examples of their 
Use and Hints on the Art of Computation. By Simon Newcomb, Pro- 
fessor of Mathematics in the Johns Hopkins University. Revised, vi r 
168 + vi + 80 + 104 pp. 8vo. $1.60. 

Elements 0/ Trigonometry separate, vi + 168 pp. Si. 20. 

Mathematical Tables, with Examples of their Use and Hints on the Art 
of Computation, vi + 80 + 104 pp. $1.10. 

The Tables, which are to five places of decimals, are regu- 
larly supplied to the United States Military Academy and to 
Princeton University and Yale University for the entrance 
examinations. 

Newcomb's Essentials of Trigonometry. 

Plane and Spherical. With Three- and Four-place Logarithmic and 
Trigonometric Tables. By Simon Nkwcomb, Professor of Mathematics 
in the Johns Hopkins University, vi + 187 pp. i2mo. $1.00. 

Much more elementary in treatment than the foregoing. 
Newcomb's Elements of Analytic Geometry. 

By Simon Newcomb, Professor of Mathematics in the Johns Hopkins 
University, viii + 357 pp. i2mo. $1.20. 

Corresponds closely to the usual college course in plane 
analytic geometry, but is so arranged that a practical course 
may be made up by omitting certain sections and adding Part 
II, which treats of geometry of three dimensions. The sec- 
tions omitted in the practical course, together with Part III, 
form an introduction to modern projective geometry. 



3 6 Mathematics 



Newcombs Elements of the Differential and Integral 
Calculus. 

By Simon Newcomb, Professor of Mathematics in the Johns Hopkins 
University, xii + 307 pp. i2mo. $1.50. 

A complete outline of the first principles of the subject 
without going into developments and applications further than 
is necessary to illustrate the principles. 

Nipher's Introduction to Graphic Algebra. 

For the use of High Schools. By Francis E. Nipher, Professor in Wash- 
ington University. i2mo. 66 pp. 60 cents. 

Eighteen of the most elementary graphs illustrating the 
solution of equations. It is thought that none of these graphs 
is beyond the capacities of high-school pupils. By injecting 
some such material here and there into the ordinary instruction 
in algebra, new meaning can be given to mathematical opera- 
tions and new interest to the whole subject. 

Phillips and Beebe's Graphic Algebra. Or Geometrical 
Interpretations of the Theory of Equations of One Un- 
known Quantity. 

By A. W. Phillips and W. Beebe, Professors of Mathematics in Yale 
College. Revised Edition, 156 pp. 8vo. $1.60. 



^ 



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